Cartan Actions of Higher Rank Abelian Groups and Their Classification

CARTAN ACTIONS OF HIGHER RANK ABELIAN GROUPS AND THEIR CLASSIFICATION

RALF SPATZIER ∗ AND KURT VINHAGE †

k ` Abstract. We study transitive R ×Z actions on arbitrary compact manifolds with a projectively dense set of Anosov elements and 1-dimensional coarse Lyapunov foliations. Such actions are called totally Cartan actions. We completely classify such actions as built from low-dimensional Anosov flows and diffeomorphisms and affine actions, verifying the Katok-Spatzier conjecture for this class. This is achieved by introducing a new tool, the action of a dynamically defined topological group describing paths in coarse Lyapunov foliations, and understanding its generators and relations. We obtain applications to the Zimmer program.

Contents

Introduction and Main Results 2 1. Introduction 2 2. Summary of Arguments7

Preliminaries, Examples and Standard Structure Theory 13 3. Dynamical Preliminaries 13 4. Topological group preliminaries 19 5. Lyapunov Metrics and Closing Lemmas 22 6. Examples of Totally Cartan Actions and Classiﬁcation of Aﬃne Actions 34

Part I. Rank One Factors and Transitivity of Hyperplane Actions 41 7. Basic properties of Lyapunov central manifolds 41 8. W H -Holonomy Returns to Lyapunov central manifolds 43 9. Finite returns: Constructing a rank one factor 48 10. Inﬁnite returns: Finding a dense H-orbit 53 11. Proofs of Theorems 2.1 and 2.2 61

Part II. Constructing a Homogeneous Structure 61 arXiv:1901.06559v3 [math.DS] 13 Sep 2020 12. Basic Structures of Geometric Brackets 62 13. Ideals of Weights 70 14. Homogeneity from Pairwise Cycle Structures 78 15. Extensions by Maximal Ideals 86 16. Proofs of Theorems 1.7 and 12.19 97

Part III. Structure of General Totally Cartan Actions 98 17. The Starkov component 98

∗ Supported in part by NSF grants DMS 1607260 and DMS 2003712. † Supported in part by NSF grant DMS 1604796. 1 18. Centralizer Structure 99 19. Virtually Self-Centralizing Actions 102 20. Applications of the Main Theorems 108 References 110

Introduction and Main Results 1. Introduction Hyperbolic actions of higher rank abelian groups are markedly different from single hyperbolic diffeomorphisms and flows and display a multitude of rigidity properties such as measure rigidity and cocycle rigidity. We refer to the survey [72] as a starting point for these topics, and [75, 16, 18,5] and [47, 12, 77] for more recent developments concerning measure and cocycle rigidity, respectively. In this paper, we concentrate on the third major rigidity property: classification and global differential rigidity of such actions. Smale already conjectured in 1967 that generic diffeomorphisms only commute with its iterates [71]. In the hyperbolic case this was proved by Palis and Yoccoz in 1989 [56, 57]. Recently, Bonatti, Crovisier and Wilkinson proved this in full generality [4]. Much refined local rigidity properties were found in the works of Hurder and then Katok and Lewis on deformation and local rigidity of the standard action of SL(n, Z) on the n-torus in which they used properties of the action of higher rank abelian subgroups [35, 43, 44]. This led to investigating such actions more systematically. Let us first coin some terminology. Given a foliation F, a diffeomorphism a acts normally hyperbolically w.r.t. F if F is invariant under a and if there exists a C0 splitting of the tangent space s u TM = Ea ⊕ T F ⊕ Ea into stable and unstable subspaces for a and the tangent space of the leaves s u of F. More precisely, we assume that Ea and Ea are contracted uniformly in either forward or backward time by a (see Definition 3.1 for precise definitions). All manifolds discussed will be assumed to be connected unless otherwise stated. This will often be superfluous, since we often assume the existence of a continuous action with a dense orbit.

k ` k ` 1 Deﬁnition 1.1. Let R × Z , and R × Z y X be a locally free C action on a compact manifold k k ` X. If a acts normally hyperbolic to the orbit foliation of R for some a ∈ R × Z , we call a an Anosov element, and α an Anosov action. If k + ` ≥ 2, the action is called higher rank. k ` k+` k+`−1 Let A ⊂ R × Z denote the set of Anosov elements, and p : R \{0} → RP denote the projection onto real projective space. We say that an action is totally Anosov if p(A) is dense in k+`−1 RP . k ` k ` An action R × Z y X is called transitive is there exists a point x ∈ X such that (R × Z ) · x is dense in X.

` Note that higher rank Anosov Z -actions correspond to ` commuting diﬀeomorphisms with at least one being Anosov. The following deﬁnition is critical to the structure theory of higher rank abelian actions.

k ` r ∞ Deﬁnition 1.2. Let k + ` ≥ 2 and R × Z y X be a C action on a C manifold X, r ≥ 1.A Cs-rank one factor, s ≤ r is • a C∞ manifold Y and a Cs submersion π : X → Y , k ` • a homomorphism p : R × Z → A, where A is a compact extension of R or Z, and s • a C action A y Y such that π(a · x) = p(a) · π(x). 2 The factor is called a Kronecker factor if the action of A is by translations of a torus.

k ` Remark 1.3. If R ×Z y X, π : X → Y is a Kronecker factor, it is not difficult to see that any two elements related by a stable or unstable manifold must be sent to the same point. In particular, ` k there are no Kronecker factors of an Anosov Z action, and the only Kronecker factors of an R action correspond to projecting onto the coordinates of the acting group. Our definition coincides with another one that is often used: one may alternatively define a Cs- 0 k l rank one factor as a factor for which a cocompact subgroup A ⊂ R × Z acts by a single flow or diffeomorphism after projecting (A0 is complementary to the A of Definition 1.2). The lack of a rank one factor can be interpreted as an irreducibility condition: many actions with rank one factors are represented as skew-products in the corresponding category where the base of the skew product is a flow. One may similarly define measurable and continuous (C0) factors of the action, in which the regularity conditions are relaxed. It is clear that every smooth rank one factor is topological, and every topological rank one factor is measurable. Forbidding smooth rank one factors is therefore weakest of the possible no rank-one factor assumptions. There are natural examples of higher rank actions coming from homogeneous actions, e.g. by the diagonal subgroup of SL(n, R) on a compact quotient SL(n, R)/Γ or actions by automorphisms of Lie groups, e.g. by commuting toral automorphisms. More generally, genuinely higher rank actions should be algebraic in the following sense (see Section6 for examples of such Anosov actions):

k l Definition 1.4. An algebraic action is an action by R × Z by compositions of translations and automorphisms on a compact homogeneous space G/Γ. The action is homogeneous if it consists only of translations. Katok and Spatzier proved local C∞ rigidity of standard actions without rank one factors in [46], generalizing the earlier works by Hurder [35, Theorem 2.19] respectively Katok and Lewis [43, Theorem 4.2]. This gives evidence for the following conjecture of Katok and Spatzier (posed as a question in [9] and as a conjecture in [33, Conjecture 16.8]). Conjecture 1.5. (Katok-Spatzier) All higher rank C∞ Anosov actions on any compact manifold without C∞ rank one factors are C∞ conjugate to an algebraic action after passing to finite covers. A Cr version of the conjecture may also hold, possibly with loss of regularity in the conjugacy. This loss of regularity occurs in many rigidity theorems for partially hyperbolic actions, see, e.g. [12]. The minimal value of r for which the conjecture should be true is still mysterious. Conjecture 1.5 is reminiscent of the longstanding conjecture by Anosov and Smale that Anosov diffeomorphisms are topologically conjugate to an automorphism of an infra-nilmanifold [71]. The higher rank and irreducibility assumptions allow for much more dramatic conclusions: First, the conjugacy is claimed to be smooth. For single Anosov diffeomorphisms, smooth rigidity results are not possible as one can change derivatives at fixed points by local changes. Furthermore, Farrell and Jones and later Farrell and Gogolev constructed Anosov diffeomorphisms on exotic tori [20, 19, 21] (although after passing to a finite cover, the exotic structure becomes standard). Thus not even the differentiable structure of the underlying manifold is determined. Secondly, the conjecture applies k k equally well to Z and R -actions. There is no version of the Anosov-Smale conjecture for Anosov flows. Even topological rigidity is out of reach as there are many flows which are not even orbit- equivalent to Anosov flows, see for instance the Handel-Thurston examples [31]. In addition, Anosov flows may not be transitive, such as the Franks-Williams examples [25]. Significant progress has been made on this conjecture in the last decade. For higher rank Anosov k Z actions on tori and nilmanifolds, Rodriguez Hertz and Wang [66] have proved the ultimate result: 3 global rigidity assuming only one Anosov element, and that its linearization does not have affine rank one factors. This improves on the results of [64], and together they are the only known results that assume the existence of only one Anosov element in the action. In fact it followed earlier work by Fisher, Kalinin and Spatzier [24] for totally Anosov actions, i.e. Anosov actions with a dense set of Anosov elements. As knowledge of the underlying manifold is required, this is really a global rigidity theorem. Finally, in the same vein, Spatzier and Yang classified nontrivially commuting expanding maps in [73]. This was possible as expanding maps were known to be C0 conjugate to endomorphisms of nilmanifolds (up to finite cover) thanks to work of Gromov and Shub [29, 69]. Of course, a positive resolution of the Anosov-Smale conjecture, combined with the results above, k would automatically prove the Katok-Spatzier conjecture for higher rank Z actions. k Much less is known for R actions or when the underlying manifold is not a torus or nilmanifold. We will concentrate on the so-called totally Cartan actions. For a totally Anosov action α, a coarse Lyapunov foliation is a foliation whose leaves are maximal (nontrivial) intersections of stable s manifolds ∩Wai for some fixed Anosov elements ai (see Section 3.1 for a thorough discussion and definition). If α preserves a measure µ of full support, this can be phrased in terms of Lyapunov exponents (see Section 3.2), thus the name. k l Definition 1.6. A (totally) Anosov action of A = R × Z , A y M, is called (totally) Cartan if all coarse Lyapunov foliations are one dimensional. We call the action Lyapunov orientable if every coarse Lyapunov foliation can be given an orientation. The notion of a Cartan action has had varying incarnations: one may insist that each coarse Lyapunov foliation is 1-dimensional and that for each coarse Lyapunov foliations W, there exists k ` u a ∈ R × Z such that W = Wa (as defined in [35]); or one may define a Cartan action to be an n−1 action of Z on an n-manifold (as defined in [43] and [38]); or we may use our definition, which coincides with the one given in [42]. It by far is the weakest of the possible definitions, is implied by all the others and there are many examples which satisfy Definition 1.6 which do not satisfy the alternatives. ∞ k Kalinin and Spatzier classified C totally Cartan actions of R for k ≥ 3 in [42] on arbitrary k manifolds under the additional hypothesis that every one-parameter subgroup of R acts transitively and α preserves an ergodic probability measure µ of full support. Later, Kalinin and Sadovskaya ` proved several strong results in this direction for totally nonsymplectic (TNS) actions of Z , i.e. ` actions for which no two Lyapunov exponents (thought of as linear functionals on Z ) are negatively proportional. They also treated higher dimensional coarse Lyapunov spaces but required additional conditions such as joint integrability of coarse Lyapunov foliations or non-resonance conditions, as well as quasi-conformality of the action on the coarse Lyapunov foliations [40, 39]. Recently, Damjanovic and Xu [14] generalized their results relaxing the quasi-conformality conditions but still requiring joint integrability or non-resonance. The joint integrability or non-resonance conditions are useful to force the coarse Lyapunov subspaces to “commute,” which guarantees that the action is on a torus. We improve on the assumptions appearing in these works in the following ways: • We establish a new method of building homogeneous structures, which allows for resonances and non-integrability, allowing for complete classification. • We do not require the existence of an ergodic invariant measure of full support (we only k ` require transitivity of the R × Z action). • We do not directly require transitivity or ergodicity for subactions (e.g. the ergodicity of one-parameter subgroups), instead deducing their transitivity from restrictions on possible rank one factors. 4 • Our conditions are purely dynamical and do not require topological assumptions or an underlying model. • We build rank one factors of the action using only dynamical input, and still obtain a classification in their presence. We begin by establishing the main result, where the action has no rank one factors. We actually prove more, allowing Kronecker factors of the action, ie factors which are translations of a torus. k This allows us to discuss the case of Z actions by passing to a their suspension (the suspension will k always have a T , and hence T, factor). We discuss suspensions and their relationship to Kronecker factors in Section 3.5. k ` 2 Theorem 1.7. Let R × Z be a C transitive, totally Cartan action, and assume no finite cover of the action has a non-Kronecker C1,θ rank one factor. Then the action is C1,θ-conjugate to an affine action (up to finite cover) for some θ ∈ (0, 1). Furthermore, if the action is C∞, and one instead assumes that there are no non-Kronecker C∞ rank one factors of finite covers of the action, then a C∞ conjugacy exists. We will use Theorem 1.7 in the proof of the full classification result. In addition, we classify all homogeneous (in fact, affine) totally Cartan actions, see Section 6.2. If one assumes that the action is Lyapunov orientable (Definition 1.6), then one may replace the assumption that every finite cover of the action has no non-Kronecker rank one factors with the assumption that the action itself has no non-Kronecker rank one factors. Therefore, in general, one only needs to check up to 2dim(X)−k-fold covers of X. Alternatively, one may assume that there are no orbifold factors of a given action. In addition to Theorem 1.7, we can completely describe the structure of totally Cartan actions, ` even with non-Kronecker rank one factors. The case of Z actions requires the least technical conditions to state, and is a corollary of the subsequent theorems. ` 2 Corollary 1.8. If α : Z y X is a C transitive, totally Cartan action. Then there exists an action m 2 r 2 Z y (T ) ×Y for some nilmanifold Y , given by the product of Anosov diffeomorphisms of T and 1,θ 2 r an affine action on Y , such that X is a C finite-to-one factor of (T ) × Y , and a finite index ` m ∞ subgroup of α(Z ) coincides with a restriction of the induced Z action. If the action is C , so is the factor map. ` k ` While the case of Z actions has simplest assumptions and conclusion, general R × Z actions are much less clean, so we instead describe them in relation to other actions. We say that an action k ` m k ` m R × Z y X embeds into an action R y Y if there exists a homomorphism σ : R × Z → R and an embedding ϕ : X → Y such that ϕ(a · x) = σ(a) · ϕ(x). This definition allows us to consider k k ` only R actions, since R × Z actions will be embedded in their suspension. We first need a few additional structural features, which we now describe. Let ∆ be the set of k coarse Lyapunov exponents of the totally Cartan action (see Section 3.2). Then let S ⊂ R be T k defined by S = β∈∆ ker β. We call S the Starkov component of the R -action. k Remark 1.9. One may alternatively describe the Starkov component as the set of a ∈ R satisfying any (and hence all) of the following conditions: (1) a has zero topological entropy. k (2) a : k ∈ Z is an equicontinuous family. (3) a does not have sensitive dependence on initial conditions. (4) a is not partially hyperbolic (with nontrivial stable and unstable bundles). Therefore, the condition that an action has trivial Starkov component is equivalent to 0 being the only element satisfying any of the above conditions. 5 k The Starkov component can be considered the “trivial” part of the R action, and we can safely k ignore it after passing to a factor. Lemma 17.2 which constructs the Starkov factor of R y X, k−` which we denote by R y X¯ = X/S. The Starkov factor carries all of the revelant hyperbolic behavior on X. Under an additional assumption, the action takes a particularly simple form. A Cr action, r ≥ 1, k 1 k R y X is called virtually self-centralizing if its C -centralizer is F × R for some finite group F . k 2 Theorem 1.10. Let R y X be a C transitive, totally Cartan action with trivial Starkov component such that every finite extension of the action is virtually self-centralizing. Then some finite cover of the action is C1,θ-conjugate to a direct product of a collection of homogeneous actions and Anosov flows on 3-manifolds. If the action is C∞, so is the conjugacy. Without the additional assumptions of Theorem 1.10 we may still prove a classification, but with a more complicated picture. The following is the most complete result of this paper: k ` 2 Theorem 1.11. Let α : R × Z y X be a C transitive, totally Cartan action. There exists a m totally Cartan, transitive action β : R y Y1 × Y2 × · · · × Yr × H/Γ such that: 1,θ t (1) Each Yi is a 3-manifold with an associated transitive C Anosov flow ϕi. m ∼ r m−r (2) If an element a ∈ R = R × R is written as a = (t1, . . . , tr; b), then:

t1 tr β(a) · (y1, . . . , yr; x) = (ϕ1 (y1), . . . , ϕr (yr); b · x) m−r where R y H/Γ is homogeneous. (3) A finite cover of the Starkov factor of α C1,θ-embeds into a finite-to-one factor of β. ∞ If the action is C , so are the Anosov flows ϕi and the embedding. Theorem 1.11 is optimal in the Cartan setting, and provides a full classification of all totally Cartan actions. It is easy to see that any action described by the conclusions of Theorem 1.11 is totally Cartan, so it is a characterization of all totally Cartan actions. Furthermore, one cannot 2 relax the totally Cartan condition to the Cartan condition: we in fact construct an example of an R action which is Cartan, but not totally Cartan (see Section 6.3.4). We furthermore cannot dispose of the condition that the Starkov factor embeds into an action β, or that one must first pass to the Starkov factor, as shown in Sections 6.3.1 and 6.3.3. As a first application, we obtain the following characterization of homogeneous totally Cartan actions: k ∞ Corollary 1.12. Let R y X be a C transitive totally Cartan action with trivial Starkov component. Then the action is homogeneous if and only if there exists an invariant volume µ and some k a ∈ R such that a is Anosov and transitive, and hµ(a) = htop(a). Corollary 1.12 is reminiscent of the Katok entropy conjecture, which states that if the Liouville measure for a geodesic flow on a negatively curved manifold is the measure of maximal entropy, then that manifold is a locally symmetric space. In the purely dynamical setting, this is known to fail, as demonstrated by the example in [27, Remark 1.3]. The failure stems from the fact that the stable and unstable manifolds have dimension larger than one, allowing Jordan blocks to appear in periodic data. Therefore, the Cartan assumption plays a crucial role by forcing diagonalizibility with simple spectrum. We can also apply Theorem 1.7 to obtain global rigidity results for totally Cartan actions on spaces of the form M = G/Γ, with G an R-split semisimple group. This result is complementary k to the global rigidity results for Anosov Z actions (for which the best results are given in [66], see the discussion above). 6 Corollary 1.13. Let G be a semisimple linear Lie group of noncompact type, Γ ⊂ G be a lattice such that Γ projects densely onto any PSL(2, R) factor, and M = G/Γ be the corresponding G- ∞ 2 k homogeneous space. Then there is at most one C (C ), transitive, totally Cartan R -action on M, up to C∞ (C1,θ) conjugacy and linear time change. Such an action exists if and only if G is R-split, and in this case, is the Weyl chamber flow on G/Γ. Theorem 1.7 also has an immediate application to the Zimmer program of classifying actions of higher rank semisimple Lie groups and their lattices on compact manifolds. We refer to Fisher’s recent surveys [22, 23] for a more extensive discussion. This program has been a main impetus for seeking rigidity results for hyperbolic actions of higher rank abelian groups. Corollary 1.14. Let G be a connected semisimple Lie group. Suppose G acts on a compact manifold M by C2 diffeomorphisms such that the restriction to a split Cartan A ⊂ G is a transitive, totally Cartan action of A. Then a finite cover of the action is C1,θ conjugate to a homogeneous action of G on a homogeneous space H/Λ via an embedding G → H. If the action is C∞, so is the conjugacy. This result represents a significant step in our understanding of semisimple group actions. Pre- viously, it had only been established under highly restrictive assumptions, in particular that G is simple, has R-rank at least 3 and that G-action preserves a volume [26]. Corollary 1.15. Let G be a semisimple Lie group such that every factor of G has rank at least 2, ∞ Γ ⊂ G be a lattice, and Γ y X be a C action of Γ. Assume that there exists an abelian subgroup A ⊂ Γ such the the restriction of the action to A is a transitive, totally Cartan action and that either a) Γ is cocompact, or b) the action of Γ preserves a probability measure. Then for some finite index subgroup Γ0 ⊂ Γ, the Γ0 action is C∞-conjugate to an affine action after passing to a finite cover. Similar theorems are known to hold under different assumptions. The main input we provide is that the space X is a nilmanifold, which allows us to quote [6]. There, the rigidity of such lattice actions is also shown to hold in other settings, with much weaker assumptions, such as the existence of an Anosov element and invariant measure, provided the action is on a nilmanifold. Other results in this direction include that of Qian in [62], where the totally Cartan condition is replaced by a condition called weakly Cartan (which neither implies, nor is implied by, the totally Cartan condition). This result also requires an invariant volume. Acknowledgements: The authors are grateful to Aaron Brown, Danijela Damjanovic, Boris Has- selblatt, Boris Kalinin, Karin Melnick, Victoria Sadovskaya and Amie Wilkinson, for their interest and various conversations, especially about normal forms. We would also like to thank David Fisher and Federico Rodriguez Hertz for comments on the paper and suggestions for improvements. We also thank University of Chicago, Pennsylvania State University and University of Michigan for support and hospitality during various trips.

2. Summary of Arguments The paper is organized into 3 parts. In part one, we discuss the relationship between transitivity conditions for subactions and the existence of rank one factors. In part two, we show that if an action satisfies a transitivity condition equivalent to the non-existence of rank one factors, then the action is homogeneous. In part three, we use the structural results of the first two parts to prove the full structure theorem. 7 2.1. Developing the Toolbox. There are four sections that appear before the main arguments and results of the paper. Many basic technical tools, especially from Anosov flows and diffeomorphisms, have been adapted from single diffeomorphisms to higher rank actions. We will use them freely, after summarizing them in Section3. In particular, we generalize notions of stable manifolds called coarse Lyapunov foliations, with associated linear functionals called weights. In the presence of an invariant measure, the weights are related to Lyapunov exponents coming from higher-rank adaptations of Pesin theory, see [7] for a detailed presentation. We also recall recent developments on normal forms, prove some folklore theorems related to the construction of Lyapunov hyperplanes and show that Anosov flows on 3-manifolds are virtually self-centralizing (Theorem 3.12). Section4 recalls results in topological groups which we will use in Part II. Section6 has several examples of actions which exhibit the features of the theorems as well as a several of the structures that we introduce, as well of a classification of affine actions. Section5 contains several of the crucial new tools we develop and use in Parts I-III. The core of the arguments throughout the paper is isometric-like behavior along coarse Lyapunov foliations and moving by their associated hyperplanes. This adapts arguments from [42] which produced a metric on coarse Lyapunov distributions such that the derivative along these distributions is given by a functional independent of x. This feature characterizes algebraic systems, and such a metric is reproduced and used in Part II. When the action is not homogeneous, one does not expect such a metric, since the derivative cocycle may not be cohomologous to a constant. However, the Lyapunov hyperplanes still have a weaker property: uniformly bounded derivatives along hyperplanes. This is proved in Lemma 5.4, and serves as an essential tool in Parts I and III. The other main result of Section5 are adaptations of classical theorems in Anosov flows to Anosov k R -actions. This includes the Anosov closing lemma (Theorem 5.8), which appeared in [47] without proof and in a more specialized setting in [1, Lemma 4.5 and Theorem 4.8]. We also prove a spectral k decomposition (Theorem 5.9). Unlike its counterpart in Anosov flows, the proof that R -periodic orbits are dense (Theorem 5.13) requires significant effort, even with the closing lemma, and is the main result of this section. The last tool we develop the geometric commutator, which is essential in understanding the way in which the coarse Lyapunov foliations interact. When the coarse Lyapunov distributions are smooth, one can take Lie brackets to answer such questions. In our case, and most dynamical cases, the vector fields corresponding to dynamical foliations are only Hölder continuous. This motivates another approach. We replace the Lie bracket with a coarser geometric version of the commutator of two foliations as W α and W β as follows: create a path by following the α and β coarse Lyapunov spaces to create a “rectangle.” We produce a canonical way to close this path up using legs from other coarse Lyapunov spaces. Those combined paths define the geometric commutator. We will often call the individual segments in a coarse Lyapunov space legs of the geometric commutator.

2.2. Part I: Rank One Factors and Transitivity of Hyperplane Actions. One of the key novelties, and difficulties, of our results is that we obtain classification results on arbitrary manifolds, without requiring irreducibility conditions, such as ergodicity of one parameter subgroups. This allows for products to appear (as described in the statement of Theorem 1.11). Indeed, since not every Anosov flow on a 3-manifold is homogeneous, we cannot hope to prove Theorem 1.7 without assumptions to rule them out. If such a rank one factor exists, by Definition 1.2, there is an associated codimension one subgroup H = ker p which acts trivially on the factor, and therefore preserves each fiber. Therefore, if the action has a rank one factor, there exists a codimension one hyperplane which does not have a dense orbit. In fact, generically, its orbit closures are exactly the fibers of the factor. 8 The converse of this observation is much more difficult to prove: one must build a factor under the assumption that hyperplane orbits are never dense. The aim of Part I is exactly this undertaking, k k ` culminating in Theorem 2.1. We restrict our attention to R -actions, and treat R × Z -actions k ` by considering their suspensions. Indeed, the relationship between rank one factors of the R × Z actions and factors of the suspension is established in Lemma 3.13.

k 2 ∞ Theorem 2.1. Let R y X be a C (C ) transitive totally Cartan action on a compact manifold k X, and fix a Lyapunov hyperplane H ⊂ R . Then either: k 1,θ ∞ • the action lifts to some finite cover R y X˜, and there is a C (C ) non-Kronecker rank one factor of the lifted action, or 2 ∞ 1 −1 • there is a C (C ) Kronecker factor onto T , and there exists x ∈ π (0) such that H · x = π−1(0), or • there exists a point x such that H · x = X. There are several important ingredients in proving Theorem 2.1. We assume throughout that H does not have a dense orbit, and aim to obtain one of the first two cases of the theorem. First, we have to build a model for the factor. We do so by first showing that if the action H is not transitive, k then among the set of R -periodic orbits, many of them are also H-periodic. Using normal forms, we are able to show that saturations of H-periodic orbits by the corresponding coarse Lyapunov foliations are actually fixed point sets for regular elements of H fixing the periodic orbit itself. We call this set M α(p). They therefore carry a canonical smooth structure, and are (k +2)-dimensional. k We further show that the restriction of the R -action to this invariant set is Cartan, and that the H-action is the Starkov component for the restricted action. Therefore, after factoring by the H-action, which is through a (k − 1)-dimensional torus, one obtains a 3-manifold onto which we factor. With the model in hand, we must define the projection. Let α denote a weight linked to H. Intuitively, in the case of a direct product, moving by the β-foliations, β 6= ±cα and H-orbits does not change the position on the factor. Therefore, it is natural to consider the sets W H (x), which are the saturations of points x by their H-orbits and β-foliations, β 6= ±cα. The main idea is then to show that W H (x) intersects the set M α(p) in a single point (or more generally, in a finite set) for every x, and defining the projection by x 7→ W H (x) ∩ M α(p). Building structure on such intersections is the main technical work of Part I. Crucially, the control of derivatives for elements of Lyapunov hyperplanes translates into control of derivatives for holonomies along coarse Lyapunov foliations, so that given a path in W H which begins and ends on the same M α(p), one can construct a holonomy map and bound its derivative based on the number of times the path “switches” between exponents. One can use this control to show that the holonomies must converge, and using normal forms that any sequence of holonomies returning to the same coarse Lyapunov leaf induces an affine map. Rigidity properties for subgroups of the affine groups allow us to conclude that W H (x) ∩ M α(p) = ∞, and with more work, that W H (x) ⊃ M α(p). In this case, we are able to show that W H (x) = X, and hence that the action of H is not transitive.

Furthermore, if the action of H is not transitive, we show that W H (x) ∩ M α(p) is independent of k x. This implies that there is a well-defined factor of the R -action, which is a finite-to-one factor of the induced Anosov flow on M α(p). 2.3. Part II: Constructing a Homogeneous Structure. In this section, we prove Theorem 1.7. The main difficulty in producing a conjugacy with a homogeneous action is building a homogeneous structure from purely dynamical, and not geometric or topological, input. Indeed, we do not assume k anything about the underlying manifold. In the case of R , k ≥ 3 actions, the construction was 9 2 carried out in [42] using a Lie algebra of vector fields, under strong assumptions. The case of R actions is significantly more difficult than that of actions of rank 3 and higher groups, and previous methods remain inadequate in our generality. Indeed, we need to develop several completely new tools. They allow us to remove several restrictive assumptions, such as ergodicity of one-parameter subgroups with respect to a fully supported invariant measure, and even the existence of such a measure at all. We expect these new ideas to be useful in other situations as well. The starting point of our argument is the main technical result about the Hölder cohomology of the derivative cocycle, which was first shown in [42, Theorem 1.2] under much stronger assumptions. We show that it also holds for totally Cartan actions of any higher rank abelian group without rank one factors. It implies immediately the following theorem, which gives a canonical homogeneous structure on each coarse Lyapunov leaf:

k ` 2 ∞ Theorem 2.2. Let R × Z y X be a C (C ) transitive totally Cartan action on a compact k l 2 ∞ manifold X. Then either the action lifts to a ﬁnite cover R × Z y X˜ and there is a C (C ) non-Kronecker rank one factor of the lifted action, or for every coarse Lyapunov foliation W, there k exists a linear functional α : R → R and a Hölder metric ||·||W on T W such that

α(a) k (2.1) ||a∗v||W = e ||v||W for all a ∈ R , v ∈ T W.

The norm ||·||W is unique up to global scalar.

The Hölder metrics make each coarse Lyapunov manifold isometric to R. After passing to a cover, we may define Hölder flows along each such manifold which act by translations in each leaf. We now discuss the main novelty of this paper: how to “glue” groups actions which parameterize only topological foliations with smooth leaves into a larger Lie group action that actually gives the total space a homogeneous structure. The method is as follows: The R-actions parameterizing coarse Lyapunov foliations allow us to define an action of the free ∗d product P := R on M where d = dim M − k. Elements of P are formal products of elements coming from each copy of R, which we call legs. Each copy of R will correspond to a flow along a corresponding coarse Lyapunov foliation. We call P the path group. With the free product topology, k P becomes a connected and path connected topological group which, when combined with the R action in a precise way, gives a transitive topological group action on M. This group P is enormous, infinite dimensional for sure, and not a Lie group (see Section4). Our main achievement is to show that this action factors through the action of a Lie group. The rough idea is to show that the cycle subgroups, the stabilizers of P at x, are normal. Such an action of P is said to have constant cycle structures, see Corollary 4.13. We will not be able to show this exactly, but instead show that every stabilizer contains a fixed normal subgroup C independent of x, for which P/C has is locally compact and for which sufficiently small subgroups act trivially (Section 14.3). Therefore, by a Theorem of Gleason and Yamabe, the action of P will factor through an action of a Lie group G. Since P acts transitively on M by construction, M will be a G-homogeneous space. Moreover, k the original action of R naturally relates to P, and becomes part of the homogeneous action. The idea of using free products to build a homogeneus structures was first explored by the second author in [76] when proving local rigidity of certain algebraic actions. Basically it is a new tool to build global homogeneous structures from partial ones on complementary subfoliations. The main insight of Part II is that these technical and seemingly narrow algebraic techniques can actually be applied to actions on any manifold, only using dynamical structures. From these observations, the main goal becomes proving constancy of the cycle relations. There are two particularly important cases: cycles consisting only of two proportional (“symplectic”) 10 weights, and commutator cycles of non-proportional weights. We call these types of cycles pairwise cycle structures (since they involve commutators of negatively proportional weights α and −cα or linearly independent weights α and β). From commutator cycles of linearly independent weights, we are able to obtain constancy of cycles whose legs belong to a stable set of weights E, where we k call E stable if for some a ∈ R , λ(a) < 0 for all λ ∈ E. Once constancy of symplectic and stable cycles is accomplished, we combine this information to prove constancy of all cycles in Section 15. In other related works on local rigidity problems, the latter was achieved via K-theoretic arguments (this first appeared in [11]). We found a new way to do this, avoiding the intricate K-theory arguments, by showing constancy of an open dense subset of relations using explicit relations between stable and unstable horocycle flows in PSL(2, R). The K-theory argument was used in the past to treat the remaining potential relations. However, density of the good relations makes this unnecessary. We first introduce a cyclic ordering of the Lyapunov hyperplanes (kernels of weights) to handle λi1 λil stable cycles (see Definition 5.30). Then we simplify products η ∗ ... ∗ η with all λij in a ti1 til stable subset E by putting the λij into cyclic order by commuting them with other λik . Assuming commutator cycles are constant we can then easily show that stable cycles are constant using the dynamics of the action, cf. Section 14.2. Combining this with the uniqueness of presentation of elements of the stable set in the order determined by a circular ordering, we get an injective continuous map from P/C. We can then apply another Lie criterion developed by Gleason and Palais (Theorem 4.12) to get a group structure on stable manifolds. To prove constancy of pairwise cycle structures coming from negatively proportional weights α, −c α for c > 0, we imitate the procedure in [26, 42], replacing their use of isometric returns with the tools of path and cycle groups. Using a point x0 with a dense ker α-orbit, we are able to take any cycle with legs in α and −c α, and obtain it as a cycle at every point in M. We can then apply the Gleason-Palais Lie criterion theorem (again) to get a Lie group structure. We use this homogeneous structure, to understand the cycles coming from α, −cα (see Lemma 12.12). The simplest case is when for all weights λ, the coarse Lyapunov foliation Wλ has a dense leaf in M. For simplicity suppose that for some α and β, we can find a λ which commutes with α, β and anything appearing in its geometric commutator. Then the commutator relation of α and β is constant along Wλ, and hence constant. A more complicated argument always works by induction, see Lemma 13.14. As simple algebraic examples show, it is not always true that actions split as a direct product of actions such that each coarse Lyapunov foliation Wλ is dense in its corresponding factor, see Section6. In such homogeneous examples, M is the total space of a smooth fiber bundle, with the corresponding tangent bundle splitting subordinate to the coarse Lypuanov splitting. We produce a topological version of this phenomenon by realizing M as a bundle over a base space B where the latter has this transitivity property for all of its coarse Lyapunov foliations. This base space is constructed by factoring out the coarse Lyapunov spaces for some suitable set of weights Emax. Crucial for constructing the base B is the following insight from Lemma 13.5: Consider a con- α tinuous function f : M → R invariant along the coarse Lyapunov foliation W for a weight α. Then for β not proportional to α, we show that f is also invariant under Wλ for any leg λ in the commutator of α and β. This is proved entirely using higher rank dynamics. We call a set of weights E an ideal if for all α ∈ E and any other weight β, all legs in the geometric commutator [α, β] belong to E (formally, this is only a consequence, but is the crucial feature that we use, see Section 13). Ideals allow us to get factors. In particular, we can factor out by a maximal ideal, and obtain a base space B (not necessarily unique). It is important to observe that B does

11 not naturally come with a differentiable structure, but that the actions and coarse Lyapunov flows desend to B. This forces us to define the notion of a topological Cartan action, which carries the data of the flows along the coarse Lyapunov foliations and the intertwining behavior, as well as other technical assumptions implied by C2-actions. We then develop the notions and ideas defined above in this setting. This allows us to conclude that on B, the remaining coarse Lyapunov spaces are all transitive, and that B will be a homogeneous space, by the arguments above and some classical results of Gleason and Yamabe (Theorem 4.11). Finally, we need to understand the fiber structure. This will be done inductively in Section 15. Roughly, we factor by a sequence of ideals of weights Ei such that each set Ei is a maximal ideal in Ei−1. This defines a sequence of factors Mi. We show that the fibers of Mi → Mi−1 are homogenous spaces, by the maximality of the ideals chosen. Then we can explore various commutator relations between weights on the base and weights on the fiber. The argument splits into base-fiber relations and base-base relations, and involves various considerations of cocycles and higher order polynomial relations.

2.4. Part III: Structure of General Totally Cartan Actions. In Part III, we prove the most general result of the paper, Theorem 1.11. We begin by establishing some structure theorems which reduce a proof of Theorem 1.11 to Theorem 1.10. In Section 17, we describe how the Starkov component acts. The uniform boundedness lemma (Lemma 5.4) will imply that the action is equicontinuous, and factors through a torus action. We then turn to the centralizers of transitive, totally Cartan actions in Section 18. In particular, we show that such centralizers act uniformly hyperbolically and are virtually self-centralizing. The difficulty is showing that the action by the centralizer is still totally Cartan. This is done by considering a weight β and an element a of the centralizer. We show that β is either isometric, uniformly expanding or uniformly contracted by a. There are two cases. In the first case, ker β has a dense orbit in which case we can show that any element of the action, and anything that commutes with it, has constant derivative independent of x. In the second case, ker β does not have a dense orbit, in which case the arguments of Part I will imply that there is a rank one factor onto which β projects as the stable or unstable manifold of some transitive Anosov flow. We may therefore lift the leafwise Margulis measures to get that every element either uniformly expands, uniformly contracts, or acts isometrically on W β. The main goal therefore turns to the proof of Theorem 1.10. The idea is as follows: apply Theorem 1.7. If there does not exist a rank one factor of the action, then we are done, since the action is homogeneous. If there does exist a rank one factor, we attempt to “peel it off” by expressing our k−1 original action as (a finite-to-one factor of) a direct product of the rank one factor with an R action on a submanifold of codimension three. If this is possible, we may repeat the process with the fiber of the rank one factor and either obtain a homogeneous structure, or peel off another rank one factor. Since the dimension is reduced by 3 each time, the process terminates, and we obtain Theorem 1.10. Much like the previous parts, the main tool here is considering paths and their induced holonomy maps. The main idea is to use the holonomy maps of the projecting nontrivially to the base to produce a section of the projection. We then apply the holonomies along the fibers to move this section along the fiber to produce a direct product structure.

2.5. Proof Locations. For convenience, the following table contains the locations of the proofs of each of the main results stated in these introductory sections: 12 Result Location Theorem 1.7 Section 16 Corollary 1.8, 1.12-1.15 Section 20 Theorem 1.10, 1.11 Section 19.2 Theorem 2.1, 2.2 Section 11

Preliminaries, Examples and Standard Structure Theory 3. Dynamical Preliminaries 3.1. Normal Hyperbolicity. Let us first recall the definition of normally hyperbolic transformations, which was used in Definition 1.1 of an Anosov action. Definition 3.1. Let M be a compact manifold with some Riemannian metric h , i. Let F be a foliation of M with C1 leaves. We call a C1 diffeomorphism a : M → M normally hyperbolic w.r.t. F if F is an a-invariant foliation and there exist constants C > 0, and λ1 > λ2 > 0 and a splitting s u s s u u of the tangent bundle TM = Ea ⊕ T Fx ⊕ Ea such that for all n ≥ 0 and v ∈ Ea, v ∈ Ea and v0 ∈ T F:

n s −λ1n s kdax(v )k ≤ Ce kv k, −n u −λ1n u kdax (v )k ≤ Ce kv k, and ±n 0 −λ2n 0 kdax (v )k ≥ Ce kv k. For the trivial foliation F(x) = {x}, a normally hyperbolic diffeomorphism w.r.t. F is just an Anosov diffeomorphism. We refer to [34, 65] for the standard facts about normally hyperbolic s u transformations. Most importantly, we obtain the existence of foliations Wa and Wa of X tangent s u r r to Ea and Ea , respectively. The foliations are Hölder with C leaves if a is C . We prove a technical lemma: Lemma 3.2. Suppose that f acts normally hyperbolically with respect to a foliation F of a manifold L X and that Vi ⊂ TX are continuous f-invariant subbundles such that TX = T F ⊕ i Vi. Then ∗ L ∗ L s u E = i(Vi ∩ E ) for ∗ = s, u, and TX = T F ⊕ i (Vi ∩ E ) ⊕ (Vi ∩ E ) . u Remark 3.3. Each intersection Vi ∩ E may be trivial, and this will happen in many examples. However, the stable and unstable bundles will be refined by taking their intersections with any Vi which is nontrivial, which is the main tool in producing coarse Lyapunov foliation.

i Proof. We claim that if πx : TxX → Vi and p : TxX → T F(x) are the projections induced by L i s s s s the splitting TxX = T F(x) ⊕ i Vi(x), then πx(E ) ⊂ E and p(E ) = 0. Indeed, if v ∈ E (x), P n n P n then v = w + i vi, where w ∈ T F(x) and vi ∈ Vi(x). Since (f )∗v = (f )∗w + i(f )∗vi must converge to 0 as n → ∞, and the subspaces remain uniformly transverse by assumption, we conclude i s s s u that πx(E ) ⊂ E and p(E ) = 0. Symmetric arguments show the same properties for E , and this s u L u immediately implies that E ⊕E = i Vi. Now, if v ∈ E (x), then according to the decomposition L TxX = T F(x) ⊕ i Vi(x),

X i X i X i v = px(v) + πx(v) = px(v) + πx(v) = πx(v). i i i i i i u i Since each πx(v) ∈ Vi by definition of πx, and πx(v) ∈ E (x) by the remarks above, πx(v) ∈ u u P u E (x) ∩ Vi(x), and E (x) = E (x) ∩ Vi(x). The sum is clearly direct, since Vi ∩ Vj = {0} for s every i 6= j. A symmetric argument follows for E (x), and the lemma follows. 13 k The following is standard for Anosov R actions. As in the introduction, we define a coarse T s Lyapunov foliation to be a foliation whose leaves are intersections W(x) = i Wai (x) of smallest positive dimension, for some fixed collection {ai} of Anosov elements.

k r Lemma 3.4. If R y X is a C Anosov action, and {ai} be a collection of Anosov elements. Then T s i Wai (x) is a Hölder foliation with smooth leaves. As an immediate corollary, we obtain:

k r Corollary 3.5. If R y X is a C Anosov action, each coarse Lyapunov foliation W is a Hölder r foliation with C leaves. The coarse Lyapunov foliations are all uniformly transverse, and TxX is k the direct sum of their tangent bundles, and the tangent bundle to the R -orbit foliations. k Proof. We prove the result by induction on the cardinality of {ai}. Let O denote the R -orbit foliation, so that the Anosov elements of the action are those which act normally hyperbolically (1) s (1) u with respect to O. Choose an Anosov element a1, and let V1 (x) = Ea1 (x) and V2 (x) = Ea1 (x), s u (1) (1) so that TxX = T O(x) ⊕ Ea1 (x) ⊕ Ea1 (x) = T O(x) ⊕ V1 (x) ⊕ V2 (x). We will proceed by induction on q ∈ N based on the following hypotheses: k • There is a chosen collection Sq = {a1, . . . , aq} ⊂ R of Anosov elements. Ln(q) (q) (q) T s • There is a splitting TxX = T O(x) ⊕ V (x) such that V (x) = E for some i=1 i i j ±acj subcollection acj ⊂ Sq. (q) r (q) (q) • There is a Hölder foliation Wi with C leaves such that TxWi (x) = Vi (x) for every i = 1, . . . , n(q). We have established the base of the induction. The induction terminates when, for every Anosov k s u element a ∈ R , Ea(x) and Ea (x) can be written as a sum of some collection of subbundles Vi(x). k s If the process has not terminated after q steps, there exists aq+1 ∈ R such E (x) cannot be written (q) as a sum of some subcollection of bundles Vi . Then according to Lemma 3.2, one may reﬁne the splitting of TxX into:

n(q) M (q) s (q) u TxX = T O(x) ⊕ Vi (x) ∩ Eaq+1 (x) ⊕ Vi (x) ∩ Eaq+1 (x) . i=1 Let V (q+1),...,V (q+1) enumerate the bundles V (q)(x) ∩ Es and V (q)(x) ∩ Eu which are 1 n(q+1) i aq+1 i aq+1 (q) (q+1) nontrivial. Since by induction, each Vi is an intersection of stable manifolds, so is Vi . Finally, (q) s we prove the foliation property. Notice that the leaves Wi (x) and Waq+1 (x) are both Hölder r (q+1) foliations with C leaves, and that the intersection of their bundles is exactly one of the Vj (x). (q+1) (q) s We claim that Wj (x) = Wi (x) ∩ Waq+1 (x) defines a foliation of X with the same properties. a (q) Indeed, given x, there exists a locally defined function ϕ : B(x, ε) → R such that Wi (x) = −1 (q) ϕ (ϕ(y)) for every y ∈ B(x, ε). Then ker ϕ∗ = V . Therefore, ϕ|W s has constant rank, since i aq+1 (q) s the kernel of its derivative at every point is exactly Vi ∩ Eaq+1 . Since it has constant rank, its preimages are C1,θ-manifolds, varying Hölder continuously. Notice that they are exactly the leaves (q+1) of Wj (x) by definition. The last thing to check is that each individual leaf is a Cr-manifold. This follows using standard (q) hyperbolic theory, and restricting attention to the bundle whose leaves are Wi (x) (which by 14 r (q) (q) s (q) u induction are C -submanifolds), with hyperbolic splitting Ei = (Ei ∩ Eaq+1 ) ⊕ (Ei ∩ Eaq+1 ) (so that the first bundle is exponentially contracting, and the second is exponentially expanding). k 1 3.2. Lyapunov Functionals and Coarse Lyapunov Spaces. Let R y X be a C action. In k the study of R actions, one often has a preferred invariant measure, such as an invariant volume. k While we make no such assumption, invariant measures for actions of R always exist, and we k discuss their properties here. If R preserves an ergodic invariant measure µ, then there are linear k k λ functionals λ : R → R and a measurable R -invariant splitting of the tangent bundle TM = ⊕E λ k such that for all 0 6= v ∈ E and a ∈ R , the Lyapunov exponent of v is λ(a). These objects may k depend on the measure µ. We call this the Oseledets splitting of TM for R , and each λ a Lyapunov functional or simply weight of the action. We let ∆ denote the collection of Lyapunov functionals. k The Lyapunov splitting is a refinement of the Oseledets splitting for any single a ∈ R . For each ¯λ L tλ λ ∈ ∆, we let E = t>0 E be the coarse Lyapunov distributions. We refer to [7] for an extensive discussion of all these topics. k For non-algebraic R -actions, the Lyapunov functionals λ may depend on the measure µ. This is typical for Anosov flows and diffeomorphisms, hence also for products of such flows. In the totally Cartan setting, we also have some additional structures, which does not assume any properties of invariant measures. Recall that a coarse Lyapunov foliation is a Hölder foliation with smooth leaves, and has associated Hölder distributions T W on M, which do not depend on the choice of measure µ. Therefore, the splitting of the tangent bundle according to Lyapunov exponents with respect to any invariant measure µ must coincide with the splitting into the coarse Lyapunov bundles. We assume that we have a fixed Riemannian metric ||·|| on M. k Lemma 3.6. If R y X is a totally Anosov action, then for each coarse Lyapunov foliation W, k there exists a unique hyperplane H ⊂ R such that: k 1 (1) for every x ∈ X, H = a ∈ R : limt→∞ t log ||(ta)∗|T W (x)|| = 0 , and k (2) if µ is an R -invariant measure, and λ is the Lyapunov functional associated to W, then ker λ = H. The hyperplane H is called the Lyapunov hyperplane corresponding to W. An example of its failure for Cartan, but not totally Cartan actions, can be found in Section 6.3.4. k 1 s u Proof. Notice that if a ∈ R is Anosov, limt→∞ t log ||d(ta)|T W || 6= 0, since T W ⊂ TW or TW . Let U denote the set of elements for which this limit is positive and V denote the set of elements for which the limit is negative. Then U ∪V contains the set of Anosov elements, and since such elements k k k are assumed to be dense (R y X is totally Anosov), U ∪ V = R . The set H = R \ (U ∪ V ) k separates R into two components. Since U and V are convex, it follows that H is a hyperplane (by the hyperplane separation theorem, see [67, Section 14.5]). Assertion (2) follows immediately. Remark 3.7. The hyperplane H appearing in Lemma 3.6 is often called a Weyl chamber wall (this terminology comes from the case of Weyl Chamber flows on semisimple Lie groups, see Section 6.1.2). k Removing all Weyl chamber walls from R , we call the connected components of the remainder the Weyl chambers of the action. For actions which are Anosov but not totally Anosov, we still call the connected components of the set of Anosov elements Weyl chambers, even though their complement may not be a union of hyperplanes. In general, each Weyl chamber is a convex set of Anosov elements, and the Anosov elements are the union of the Weyl chambers. Definition 3.8. Let ∆ denote a set of functionals such that for every coarse Lyapunov foliation W, there exists a unique functional α ∈ ∆ such that H = ker α and any a for which α(a) > 0 expands k W. An element a ∈ R will be called regular if α(a) 6= 0 for all α ∈ ∆. 15 k It is clear that the set of regular elements of R is open and dense. Remark 3.9. The functionals α are defined only up to positive scalar multiple in this context. Therefore, without loss of generality, we may assume that if there are Lyapunov foliations whose corresponding half-spaces are opposites, that their corresponding functionals are exact opposites, α and −α (this simplifies notation). In Part II, the precise exponents will exist and we will need them in our analysis, so we will not be able to make this assumption, see Remark II.2.

3.3. Normal Forms Coordinates. Let f : X → X be a uniformly continuous transformation of a compact space X, L = X × R be the trivial line bundle over f, and F : L → L be a r uniformly contracting C extension of f. That is, F takes the form F (x, t) = (f(x),Fx(t)), where r r 0 x 7→ Fx ∈ C (R, R) is continuous in the C topology and ||Fx(t)|| < 1 − ε for all (x, t) ∈ L.A r C -normal form coordinate system for F is a map H : L → L of the form H(x, t) = (x, ψx(t)) such that: r (1) Hx ∈ C (R, R) for every x ∈ X, 0 (2) Hx(0) = 0 and ψx(0) = 1 for every x ∈ X, r (3) x 7→ ψx is continuous from X to C (R, R), and (4) HF (x, t) = FHˆ (x, t), ˆ 0 where F (x, t) = (f(x),Fx(0)t). Katok and Lewis were the ﬁrst to construct normal forms on contracting foliations in the 1-dimensional setting for C∞ transformations [43]. Guysinsky extended this to the Cr setting [30]. The theory of normal forms has since been extended to a much more general setting. The broadest results in the uniformly hyperbolic setting have been written by Kalinin in [37] and even extended to the nonuniformly hyperbolic setting by Melnick [52] and Kalinin and Sadvoskaya [41].

Theorem 3.10. Let F be a uniformly contracting Cr extension of a Cr diﬀeomorphism f, r > 1. (a) F has a unique Cr-normal form coordinate system, varying Hölder continuously in the Cr- topology. (b) Any C1 normal form coordinate system coincides with the Cr system. (c) If g : X → X is a continuous transformation commuting with f, with a Cr extension G commuting with F , then G is also linear in the Cr-normal form coordinate system.

Let f be a diffeomorphism preserving a 1-dimensional foliation Wα with Cr leaves varying r r ¯α α continuously in the C topology. Fix any family of C parameterizations ψx : R → W (x) of the leaves of W α, varying continuously in the Cr topology. Then there is a corresponding r C extension of the action by defining F (x, t) = (f(x), s), where s ∈ R is uniquely defined by ¯α ¯α α α f(ψx (t)) = ψf(x)(s). Then let ψx : R → W (x) be the normal forms coordinates as guaranteed α α 0 above, so that: f(ψx (t)) = ψf(x)(f (x)t).

3.4. Centralizers of transitive Anosov flows in dimension 3. We will build rank-one factors that are Anosov flows on 3-manifolds. Understanding their properties will be important to our analysis. The following lemma is folklore in the study of Anosov flows, and the existence of the family is proven in the classical work of Margulis, see [32, 51]. The family satisfying these properties is used to construct the measure of maximal entropy, which has by now many alternate constructions. We could not find a proof of uniqueness for the family below, which we offer here for completeness.

u Lemma 3.11. Let (gt) be a transitive Anosov flow on a compact manifold Y , and {W (y): y ∈ Y } be the unstable foliation of (gt). Then 16 u u (1) there exists a continuously varying family of σ-finite, Borel measures µy defined on W (y), which are finite on compact subsets in the leaf topology, (2) the measures satisfy

(3.1) (g ) µu = ethµu , t ∗ y gt(y)

where h is the topological entropy of (gt), and s cu cu cu cu (3) if z ∈ W (y), and π : W (y) → W (z) is the stable holonomy map, then π∗µy = µz , cu cu u where µy is the measure on W (y) obtained by integrating the family µy . u Furthermore, the family µy is determined up to global scalar by these properties. u Proof of Uniqueness. We wish to show that properties (1)-(3) characterized the family µy up to a u global scalar. Suppose that νy is another such family. Then following the Margulis construction, one can construct measures µ and ν on Y , whose local disintegrations along Sinai partitions (whose atoms are generically open subsets of unstable manifolds) are absolutely continuous with respect to the respective families. Notice also that in both cases, hµ(gt) = hν(gt) = htop(gt). This implies u that the disintegrations along Sinai partitions coincide, which in particular implies that µy and u νy are absolutely continuous with respect to each other. Let f : Y → R be the Radon-Nikodym u f u derivative µy = e · νy defined µ-almost everywhere. Then one can easily check that f is invariant u u u u under gt by (3.1) for both µy and νy . Therefore, µy = λνy for a fixed constant λ on a set of full µ-measure by ergodicity with respect to µ. Therefore, since the measure of maximal entropy is fully u u supported for any transitive Anosov flow and the families µy and νy vary continuously, they agree everywhere up to a global constant. We will use the following result, which may be of independent interest. It requires low dimension, since it is easy to construct examples of Anosov flows on manifolds of dimension greater than 3 which have infinite index in their centralizers (for instance, by suspending certain Anosov automorphisms d of T , d ≥ 3).

Theorem 3.12. Let (gt) be a transitive Anosov ﬂow on a compact 3-dimensional manifold Y , and CHomeo(Y )(gt) denote the group of homeomorphisms that commute with (gt). Then CHomeo(Y )(gt) = {gt ◦ f : t ∈ R, f ∈ F } for some ﬁnite group F ⊂ Homeo(Y ).

Proof. Fix some φ ∈ Homeo(Y ) that commutes with (gt). Since (gt) is a transitive Anosov flow, it W u has a unique measure of maximal entropy, µ. Let µx denote the family of conditional measures in Lemma 3.11. Note that φ must preserve the Margulis measure of (gt), since φ∗µ is also a measure of maximal entropy for (gt). It also preserves the stable and unstable foliations of (gt) and Lebesgue measure W u W s on the orbits of (gt). This implies that φ must also act on the measures µx and µx . Since the W u Margulis measure has local product structure, and φ preserves the Margulis measures, φ∗µx = W u W s −1 W s λxµφ(x) and φ∗µx = λx µφ(x) for some fixed λx ∈ R. Notice also that λx varies continuously W ∗ since the measures µx do and is invariant under the flow, so it must be constant by transitivity. Therefore, we may find a time t of the flow such that g−t ◦ φ preserves the Margulis measure, and the leaf-wise Margulis measures. W u W u W s W s We claim that F = f ∈ Homeo(Y ): f∗µ = µ, f∗µ = µ , and f∗µ = µ is a finite u s group. Notice that one may use the leafwise measures to define a length on Wy and Wy for every y ∈ Y . Then define a length metric δ on Y by declaring the length of a broken path in u s the (gt)-orbit, W and W -foliations to be the sum of the leafwise-measures of each leg. Since the 17 Margulis measures are fully supported, the new metrics defined on the stable and unstable foliations induce the same topology on them, and δ induces the same topology on Y . Hence, (Y, δ) is a compact metric space, so Isom(Y, δ) is a compact group [15]. Finally, we show that F := Isom(Y, δ) ∩ CHomeo(Y )(gt) is discrete, which is sufficient since any discrete, compact group is finite. Let φ ∈ F be C0-close to the identity, so that φ(x) always belong to a neighborhood with local product structure. Notice that since φ commutes with (gt) d(gt(φ(x)), gt(x)) = d(φ(gt(x)), gt(x)) is uniformly bounded for all t ∈ R. This implies that φ(x) ∈ cs cu W (x) ∩ W (x), so that φ fixes every orbit of gt. Therefore, there exists a continuous function h : Y → R such that φ(x) = gh(x)(x). Since φ commutes with gt, h(gt(x)) = h(x). Therefore h is constant since gt is transitive. Therefore, φ is a time-t map of the flow. By (3.1), since φ ∈ F , we conclude that φ is the identity.

k 3.5. Rank one factors of Z -actions and their suspensions. We recall the suspension con- k k ` k ` struction of a Z action, or more generally an R × Z action. Let R × Z y X be a smooth ` or continuous action, and let X¯ = X × R . Introduce an equivalence relation ∼ on X¯, where ` (x1, a1) ∼ (x2, a2) if and only if there exists a ∈ Z such that a · x2 = x1 and a2 = a1 + a. The k ` space X˜ = X/¯ ∼ is called the suspension space of R × Z y X. A fundamental domain for this equivalence relation is X × [0, 1]`. Notice that if k = 0 and ` = 1, this is the standard suspension construction for the diﬀeomorphism generating Z. The general construction is a higher-rank version of this. Notice that the map x 7→ (x, 0) is an embedding of X into X˜, and also embeds the action by construction. Furthermore, the space X˜ carries a canonical Cr structure if the action is Cr, since the relation ∼ will be given by Cr diﬀeomorphisms. ` ` ` T is always a factor of the suspension action, since one may project onto the R /Z component, and one may therefore factor onto R/Z and obtain a rank one factor of the action, which is transitive (in the group theoretic sense). It therefore makes sense to allow Kronecker factors. The following Lemma does not require any hyperbolicity assumptions:

k+` k ` r Lemma 3.13. Let R y X˜ be the suspension of a transitive action R × Z y X by C dif- k+` r feomorphisms, r ≥ 1. Then R y X˜ has a C non-Kronecker rank one factor if and only if k ` r R × Z y X has a C non-Kronecker rank one factor. k+` Proof. Let π : X˜ → Y be a non-Kronecker rank one factor of the suspension action R y X˜ k ` ` of R × Z y X. Recall that X˜ is a fiber bundle over T , whose fibers are copies of X. Let ˜ ` ` X0 ⊂ X be the fiber above 0 ∈ T , and Y0 = π(X0). Similarly, if Xt is the fiber above t ∈ R , ` let Yt = π(Xt) = t · Y0. If Y0 is a singleton, so is Yt. Hence the R action on Y is a factor of the ` ` translation action R y T , and must be Kronecker. Therefore, Y0 is nontrivial. ` We claim that Y0 must be a smooth submanifold of Y . Indeed, let B = t ∈ R : t · Y0 = Y0 , and k+` ` ` ` B0 = t ∈ R : t · Y0 = Y0 , so that B = B0 ∩ R . Then B is a closed subgroup of R , and R /B is a factor of Y . The factor map is given by sending Yt to t (mod B). Now, choose any subgroup ◦ ` transverse to B , the connected component of B, in R , call it V . Then the action of V on Y is locally free, since it must locally permute the fibers. Therefore, at every point, the projection map ` ` Y → R /B has full rank (it takes V onto the tangent space to R /B). This implies that Y0, which ` r is the inverse image of 0 under Y → R /B, is a smooth submanifold, and π : X = X0 → Y0 is a C map. k ` Finally, we claim that π|X0 : X0 = X → Y0 is a non-Kronecker rank one factor of R × Z y X. k ` ˜ Indeed, R × Z acts on both X0 and Y0. Since Y is a rank one factor of the action on X, the stabilizer of every point contains a codimension one hyperplane, so B0 contains a codimension k ` hyperplane. In fact, this hyperplane fixes Y0 pointwise. Notice that since R ×Z is transitive, there 18 k ` is a dense orbit of R × Z on X, and so the orbit of the corresponding point in X0 has a dense k+` ˜ k+` k+` R orbit in X. The rank one factor is non-Kronecker, so B0 6= R , any dense R orbit would ` ` map to a single point, and the factor would be a singleton. Since B ⊃ Z , the Z -component of the k+` hyperplane which acts trivially must be rational (since it is codimension one, but not all of R ). k ` `−1 Therefore, either R acts trivially, and some subgroup of Z isomorphic to Z acts trivially on Y0, ` or R maps trivially. That is, the induced action on Y0 is a rank one factor. The following can be thought of as a converse of Lemma 3.13, allowing us to claim that the existence of Kronecker rank one factors is equivalent to being conjugate to the suspension of an action with larger discrete component of the acting group. We do not use this result directly, but provide it to add additional structure to actions with Kronecker factors.

k ` r Lemma 3.14. Let R × Z y X be a C transitive, totally Cartan action for r = 2 or ∞, and r π : X → T be a C Kronecker rank one factor. Then there exists a transitive, totally Cartan action k−1 `+1 r k ` −1 −1 R × Z y Y which is C -conjugate to the action of A = a ∈ R × Z : a · π (0) = π (0) on π−1(0).

−1 Proof. Consider the set Y = π (0) ⊂ X. Since π : X → T is a factor of the action, there is an k associated homomorphism σ : R → R such that π(a · x) = σ(a) + π(x). If X is the generator of a one-parameter subgroup transverse to ker σ, we may conclude that π∗X is nowhere vanishing, so π −1 r k ` is a submersion. Hence, π (0) is a C submanifold. Furthermore, there exists a periodic R × Z orbit by the Anosov Closing Lemma (one must first pass to a suspension, see Theorem 5.8), so there ` exists a point p ∈ X such that a finite index subgroup of Z fixes p. This implies that a finite index ` −1 k−1 `+1 subgroup of Z fixes π (0), and hence A is isomorphic to R × Z . Finally, it is clear the the −1 action of A on π (0) must be transitive, and totally Cartan.

4. Topological group preliminaries

4.1. Free products of topological groups. Let U1,...,Ur be topological groups. The topological free product of the Ui, denoted P = U1 ∗ · · · ∗ Ur is a topological group whose underlying group structure is exactly the usual free product of groups. That is, elements of P are given by

(i1) (iN ) u1 ∗ · · · ∗ uN where each ik ∈ {1, . . . , r} and each uk ∈ Uik . We call the sequence (i1, . . . , ik) the combinatorial (ik) pattern of the word. Each term uk is also called a leg and each word is also called a path. This is because in the case of a free product of connected Lie groups, the word can be represented by a (i1) (i1) (i2) path beginning at e, moving to u1 , then to u1 ∗ u2 , and so on through the truncations of the word. The multiplication is given by concatenation of words, and the only group relations are given by

(4.1) u(i) ∗ v(i) = (uv)(i) (4.2) e(i) = e ∈ P

Notice that the relations (4.1) and (4.2) give rise to canonical embeddings of each Ui into P. We therefore identify each Ui with its embedded copy in P. The usual free product is characterized by a universal property: given a group H and any collection of homomorphisms ϕi : Ui → H, there exists a unique homomorphism Φ: P → H such that Φ|Ui = ϕi. The group topology on P may be similarly deﬁned by a universal property, as ﬁrst proved by Graev [28]: 19 Proposition 4.1. There exists a unique topology τ on P (called the free product topology) such that

(1) each inclusion Ui ,→ P is a homeomorphism onto its image, and (2) if ϕi : Ui → H are continuous group homomorphisms to a topological group H, then the unique extension Φ is continuous with respect to τ.

In the case when each Ui is a Lie group (or more generally, a CW-complex), Ordman found a more constructive description of the topology [54]. Indeed, the free product of Lie groups is covered by a disjoint union of combinatorial cells.

Deﬁnition 4.2. Let P be the free products of groups Uβ, where the β ranges over some indexing set ∆.A combinatorial pattern in ∆ is a ﬁnite sequence β¯ = (β1, . . . , βn) such that βi ∈ ∆ ¯ for i = 1, . . . , n. For each combinatorial pattern β, there is an associated combinatorial cell Cβ¯ =

Uβ1 ×· · ·×Uβn . If each Uβ is a topological group, Cβ¯ carries the product topology from the topologies (β1) (βN ) on Uβi . Notice that each Cβ¯ has a map πβ¯ : Cβ¯ → P given by (u1, . . . , uN ) 7→ u1 ∗ · · · ∗ uN . F Furthermore, if C = β¯ Cβ¯ and π : C → P is deﬁned by setting π(x) = πβ¯(x) when x ∈ Cβ¯, then π is onto.

Lemma 4.3 ([76] Proposition 4.2). If each Ui is a Lie group, τ is the quotient topology on P induced by π. In particular, f : P → Z is a continuous function to a topological space Z if and only if its ¯ pullback f ◦ πβ¯ to Cβ¯ is continuous for every combinatorial pattern β.

Corollary 4.4. If each Ui is a Lie group, P is path-connected and locally path-connected.

∗r 1 r Let P = R be the r-fold free product of R. Given continuous ﬂows η , . . . , η on a space X, we may induce a continuous action η˜ of P on X by setting:

η˜(t(i1) ∗ . . . t(in))(x) = ηi1 ◦ ... ◦ ηin (x) 1 n t1 tn This can be observed to be an action of P immediately, and continuity can be checked with i either the universal property (considering each η as a continuous function from R to Homeo(X)) (i1) (iN ) or directly using the criterion of Lemma 4.3. Given a word t1 ∗ . . . tN (which we often call a path as discussed above), we may associate a path in X deﬁned by:

s + k − 1 ik γ = η (xk−1), s ∈ [0, 1], k = 1, . . . , m, m stk where x is a base point and x = ηik (x ). This gives more justiﬁcation for calling each term t(ik) 0 k tk k−1 k a leg. The points xk are called the break points or switches of the path.

Remark 4.5. Given a collection of oriented one-dimensional foliations Wβ1 ,...,W βN and some i βi Riemannian metric on X, one can define flows ηt according to unit speed flow along W . Therefore, even if there is no distinguished Riemannian metric to use, we will often use the terms paths, legs and break points in the foliations Wβi (especially in Part I). The combinatorial patterns of such paths still make sense even without reference to a fixed parameterization.

n Given λ = (λ1, . . . , λn) ∈ R , let ψλ denote the automorphism of P deﬁned in the following way. (i1) (im) Suppose that t1 ∗ · · · ∗ tm ∈ P is an element of combinatorial length m. Then deﬁne:

(i1) (i2) (im) (i1) (i2) (im) t1 ∗ t2 ∗ · · · ∗ tm 7→ (λi1 t1) ∗ (λi2 t2) ∗ · · · ∗ (λim tm) . 20 k ∗ k Definition 4.6. Fix a finite collection ∆ = {α1, . . . , αn} ⊂ (R ) , and let a ∈ R . Define ψa = ˆ k ψ(eα1(a),...,eαn(a)), and P = R n P, with the semidirect product structure given by

−1 (a1, ρ1) · (a2, ρ2) = (a1a2, ψa2 (ρ1) ∗ ρ2) k where the group operation of R is written multiplicatively. Proposition 4.7. Let C be a closed, normal subgroup, and H = P/C be the corresponding topological k group factor of P. If ψa(C) = C for all a ∈ R , then ψa descends to a continuous homomorphism ψ¯a of H. Furthermore, if H is a Lie group with Lie algebra h, then ¯ (1) each generating copy of R in P projects to an eigenspace for dψa on H, (2) if the ith and jth copies of R do not commute, the Lie algebra commutator of their generators, α (a)+α (a) [Xi,Xj] in H is an eigenspace of ψ¯a with eigenvalue e i j , and (3) if Y = [Z1, [Z2,..., [ZN ,Z0] ... ]], with Zk = Xi or Xj for every k, then Y is an eigenvector ¯ uαi(a)+vαj (a) of ψa with eigenvalue of the form e with u, v ∈ Z+.

Proof. If h = ρC is an element of the quotient group H, deﬁne ψ¯a(h) = ψa(ρC) = ψa(ρ)C. Let π : P → H denote the projection from P to H. For each i, let fi : R → P denote the inclusion of R th into the i copy of R generating P. Then π ◦ fi : R → H is a one-parameter subgroup of H, and we denote its corresponding Lie algebra element by Xi. Observe that since ψ¯a ◦ π ◦ fi(t) = π ◦ ψa ◦ fi = α (a) α (a) π ◦ fi(e i t), and hence dψ¯a(Xi) = e i Xi, proving (1). To see (2), observe that

αi(a) αj (a) αi(a)+αj (a) dψ¯a[Xi,Xj] = [dψ¯aXi, dψ¯aXj] = [e Xi, e Xj] = e [Xi,Xj] A similar argument shows (3). 4.2. Hölder transitivity of generating subgroups. In this section, we prove a folklore theorem about generating subgroups of a Lie group G. This property was used in [11], and follows from arguments of transitivity of foliations. A similar statement also holds for a family of foliations such that the sum of their distributions is everywhere transverse and totally non-integrable.

Deﬁnition 4.8. A family F1,..., Fn of foliations on a manifold X with metric dX are called locally θ-Hölder transitive if for every x ∈ X, there exists an > 0 and C > 0 such that if d(y, x) < , there are points x = x0, x1, . . . , xk = y such that:

(1) xi+1 ∈ Fmi (xi) for some 1 ≤ mi ≤ n P θ (2) d(xi, xi+1) < d(x, y)

Lemma 4.9. If U1,...,Un ⊂ G are generating subgroups of a Lie group G, then the coset foliations of Ui are θ-Hölder transitive for some θ > 0.

Proof. Pick vectors V1,...,Vm ∈ Lie(G) such that each Vi ∈ Lie(Uki ) for some ki and the Vi generate Lie(G) as a Lie algebra. Then there are ﬁnitely many W1,...,W` such that each Wj is an iterated bracket of the elements of {Vi} and such that {Vi,Wj} generate Lie(G) as a vector space.

If Wj = [Vi0 , [Vi1 ,..., [Viq−1 ,Viq ],..., ]], let

1/2 1/4 1/2q 1/2q ϕj(t) = [exp(t Vi0 ), [exp(t Vi1 ),..., [exp(t Viq−1 ), exp(t Viq )],..., ]] The brackets in the deﬁnition of ϕj are commutators in the Lie group, not the Lie algebra. Notice 0 that ϕj(t) satisﬁes ϕj(0) = Wj. Therefore, the map:

(s1, . . . , sm, t1, . . . , t`) 7→ exp(s1V1) ... exp(smVm)ϕ1(t1) . . . ϕn(t`) 21 has full-rank derivative at 0 and is onto a neighborhood of e ∈ G. If the number of required brackets −q to express each Qj is ≤ q, this gives 2 -Hölder transitivity. Lemma 4.9 immediately implies the following useful result.

Lemma 4.10. Suppose that a Lie group G is generated by subgroups U1,...,Un, and that η : G y X is an action of G on a compact metric space. If the restriction of η to each subgroup Ui is locally Lipschitz, then η is a locally Hölder action. 4.3. Lie Criteria. In this subsection, we recall two deep results for Lie criteria of topological groups. The first was obtained by Gleason and Yamabe [74, Proposition 6.0.11]: Theorem 4.11 (Gleason-Yamabe). Let G be a locally compact group. Then there exists an open subgroup G0 ⊂ G such that, for any open neighborhood U of the identity in G0 there exists a compact normal subgroup K ⊂ U ⊂ G0 such that G0/K is isomorphic to a Lie group. Furthermore, if G is connected, G0 = G. Recall that a locally compact group G has the no small subgroups property if for G0 as in Theorem 4.11, a small enough neighborhood U ⊂ G0 as above, U does not contain any compact normal subgroup besides {1}. Such a group G then is automatically a Lie group, by Theorem 4.11. The second criterion is obtained by Gleason and Palais: Theorem 4.12 (Gleason-Palais). If G is a locally path-connected topological group which admits an injective continuous map from a neighborhood of G into a finite-dimensional topological space, then G is a Lie group. Theorem 4.12 has an immediate corollary for actions of the path group P defined in Section4:

Corollary 4.13. If η : P y X is a group action on a topological space X, and there exists x0 ∈ X such that C := Stabη(x0) ⊂ Stabη(x) for every x ∈ X, then C is normal and the η action descends to P/C. If there is an injective continuous map from P/C to a ﬁnite-dimensional space Y , then P/C is a Lie group.

Proof. We ﬁrst show that C is normal. Let σ ∈ C and ρ ∈ P. Then σ · (ρ · x0) = ρ · x0, since σ −1 −1 stabilizes every point of X. Therefore, ρ σρ · x0 = x0, and ρ σρ ∈ C, and C is a closed normal subgroup. By Corollary 4.4, P, and hence all of its factors, are locally path-connected. Therefore, by Theorem 4.12, P/C is a Lie group if it admits an injective continuous map to a ﬁnite-dimensional space.

5. Lyapunov Metrics and Closing Lemmas 5.1. Equicontinuity of Lyapunov hyperplanes. The following is not stated explicitly in [42], but follows from equation (1) and the related discussion in the proof of Proposition 3.1 of that paper. The argument requires certain closing lemmas, and other dynamical estimates involving Hölder regularity of distributions. Crucially, the proof of the proposition does not require the assumptions of the main theorem of that paper, but only the totally Cartan assumption. The arguments work verbatim, with the exception of a normal forms result (Lemma 4.4 of [42]) which needs a low regularity version (Theorem 3.10). One may also notice that a reference to a paper of Qian [60] can be replaced with our proof of the Closing lemma (Theorem 5.8). Proposition 5.1 (Kalinin-Spatzier, [42]). Let W α be a coarse Lyapunov foliation with Lyapunov hyperplane H. For every suﬃciently small β > 0 (depending on the Hölder regularity of the coarse 22 Lyapunov foliations), there exists ε0 > 0 such that if a ∈ H is suﬃciently far from each other Lyapunov hyperplane and x ∈ X satisfy d(x, a · x) < ε0, then β (5.1) ||a∗|TW α (x)|| − 1 < d(x, a · x) . Proposition 5.1 combined with the following lemma are the main inputs for the Lemma 5.4, which plays a crucial role in Parts I and III of this paper. It establishes equicontinuity of hyperplane actions along their corresponding coarse Lyapunov foliations.

Lemma 5.2. Let X be a compact metric space X, f : X → X be a homeomorphism, and ϕ : X → R be continuous. Assume there exists some δ, ε > 0 such that if d(f n(x), x) < δ,

n−1 X (5.2) ϕ(f i(x)) < ε. i=0 PN−1 i Then there exists A such that i=0 ϕ(f (x)) ≤ A for all N ∈ N.

Proof. Fix N ∈ N. We deﬁne a sequence of times 0 = m0 < n1 < m1 < n2 < m2 < . . . depending on N, and which will terminate when either some nk = N or mk = N. Deﬁne:

n m n1 = inf ({n > 0 : ∃m such that n < m ≤ N and d(f (x), f (x)) < δ} ∪ {N}) . n m Notice that if 0 ≤ m < n1 and n ∈ [0,N] \{m}, then B(f (x), δ/2) ∩ B(f (x), δ/2) = ∅. If n1 = N, we terminate the sequence, otherwise we set

n1 m m1 = sup {n1 < m ≤ N : d(f (x), f (x)) < δ} .

As n1 6= N, then m1 is well deﬁned by deﬁnition, and at most N. Inductively, we set:

n m ni+1 = inf ({mi < n : ∃m > n such that d(f (x), f (x)) < δ} ∪ {N}) , and

ni+1 m mi+1 = sup {ni+1 < m ≤ N : d(f , f (x)) < δ} . S Notice that, similarly to the observation after the definition of n1, if `1, `2 ∈ i[mi, ni+1), then B(f `1 (x), δ/2) ∩ B(f `2 (x), δ/2) = ∅. Since X is compact, the cardinality of a δ/2-separated set is P bounded by some S. Therefore, ni+1 − mi < S. We also claim that {f ni (x)} is δ/2-separated. Indeed, if B(f ni (x), δ/2) ∩ B(f nj (x), δ/2) 6= ∅, then d(f ni (x), f nj (x)) < δ. Without loss of generality assume i < j. This contradicts the choice of mi+1, since ni < mi+1 < nj. Therefore, there are at most S intervals of the form [ni, mi]. We break the Birkhoff sum over the first N iterates into intervals (mi, ni+1) and [ni, mi]. If λ = max ϕ, then we get that:

N−1 X ϕ(f i(x)) ≤ S(ε + λ) i=0 where we bound each sum Pmi ϕ(f j(x)) using (5.2). j=ni Remark 5.3. One may notice some similarities between our proof and a proof of the Birkhoff ergodic theorem. This is perhaps not surprising, since one obtains that the Birkhoff average of ϕ tends to 0 at least linearly as a consequence of the statement, proving the Birkhoff ergodic theorem for any invariant measure of f. One may also notice that the output of Lemma 5.2 is one of the assumptions of the Gottschalk-Hedlund theorem, together with the assumption that the system is 23 minimal. While in our application, the map f will not be minimal, we note that this allows for a weakening of the uniform boundedness assumption.

k 1,α Lemma 5.4. Let R y X be a C totally Cartan action and W be a coarse Lyapunov foliation with coarse Lyapunov hyperplane H. Then there exists A > 0 such that for all h ∈ H the derivatives kh∗|T W k < A are uniformly bounded on X independent of the base point and h ∈ H. Proof. It suffices to show this for one-parameter subgroups as finitely many one-parameter subgroups generate H. We may furthermore assume that the one-parameter subgroups are far from the other Weyl chamber walls, so that Proposition 5.1 applies. Fix any h ∈ H generating such a subgroup, and notice that by fixing ε > 0, one obtains (5.2) for log ||h∗|T W || from Proposition 5.1. Therefore, we may conclude that log ||(nh)∗|T W || is uniformly bounded for all n ∈ Z by Lemma 5.2. Since log ||(th)∗|T W || is uniformly bounded for t ∈ [0, 1] and R = Z + [0, 1], we get that log ||(th)∗|T W || is uniformly bounded for all t ∈ R, as claimed. α Recall the normal forms for contracting foliations described in Section 3.3, and let ψx be the normal forms chart for W α(x) at x. Lemma 5.5. Suppose that x ∈ X, W α is a coarse Lyapunov foliation, and H is the corresponding Lyapunov hyperplane. Fix any Riemannian metric on X. Then α (a) if bn is a sequence of elements in H such that bn · x → x, and y ∈ W (x), then bn · y → y, and 0 α (b) if bn is a sequence of elements in H such that bn · x → x ∈ W (x), then after passing to a α −1 α subsequence, (ψx ) ◦ lim bn ◦ ψx is a translation.

Proof. For (a), according to Proposition 5.1, the derivative of bn restricted to the α foliation must converge to 1. Therefore, by Theorem 3.10, the corresponding linear map describing the normal forms must converge to Id. Since the normal forms and leaves of W α vary continuously, we get that α bn · y → y for all y ∈ W (x). For (b), notice that by Lemma 5.4, the maps (ψα )−1 ◦ b ◦ ψα converge to a linear map bn·x n x α α 0 f0 : R → R such that lim bn · ψx (t) = ψx0 (f0(t)). Since the normal forms charts at x and x are α −1 α related by an affine transformation, we conclude that f(t) = (ψx ) ◦ lim bn ◦ ψx is affine. Write f(t) = mx + b, and suppose that m 6= 1. Then f has a unique fixed point on R call it z, and by definition, bn · z → z. By part (a), this implies that f = Id, so m = 1. Since we assumed that m 6= 1, this is a contradiction, and f is a translation. Remark 5.6. Throughout the remainder of Section5, we will prove several features of totally Cartan action. When the action has no rank one factor, we will see that Lemma 5.4 can be strengthened to yield isometric behavior. We axiomatize the features of such actions later in Definition 12.2, which gives a remarkable rigidity result in the C0 category. The results below only use the axioms listed there, and we invite the reader to check this while reading to prevent the need to read twice. 5.2. Closing lemmas. The main result of this section will be familiar to those fluent in Anosov flows. We prove a higher-rank version of the Anosov closing lemma, which has been used in [47] without proof. A proof appeared in [1, Lemma 4.5 and Theorem 4.8], but was in a more geometric setting.

k k Definition 5.7. If R y X is a locally free action, we say that x ∈ X is R -periodic if StabRk (x) k k is a lattice in R . We call StabRk (x) the period of x. If a ∈ R , we say that a sequence of points xi and times ti are an (a, ε)-pseudo-orbit if d((tia) · xi, xi+1) < ε. 24 The Anosov closing lemma will require one additional assumption compared to the rank one setting. Indeed, consider the case of a product of two Anosov flows: Y = X1 × X2. Then there are two Lyapunov exponents, which are dual to the standard basis, call them α and β. If (p, x) ∈ Y is a ∼ point such that p is periodic and x is not, then StabR2 (p, x) = Z(t0, 0) for some t0 ∈ R. That is, the returns to (p, x) occur along the Weyl chamber wall ker β. While the first factor of Y can be made periodic, we have no control over the behavior of the second coordinate. We show that returning 2 near the Weyl chamber walls is the only potential obstacle to finding a nearby R -periodic orbit. k k Theorem 5.8 (Anosov Closing Lemma). Let R y X be an Anosov action, a0 ∈ R be an Anosov element belonging to the set of elements satisfying:

λ||a|| −λ||a|| u s (5.3) da|Ea (x) > e and da|Ea (x) < e . k Let C ⊂ R be the open cone

n k n o (5.4) C = a ∈ R : a satisﬁes (5.3) for some n ∈ N

n k o ∩ a ∈ R : d(a, ta0) < t/(10λ) for some t ∈ R so that C \{0} consists of Anosov elements. Then there exists ε > 0 and K > 0 such that if n 0 {(xi, ti)}i=1 is an (a, ε)-pseudo-orbit and d(x1, (tna) · xn) < ε, then there exists x ∈ X such that 0 k 0 x is R -periodic and d(x , x1) < Kε. If there is only one orbit segment of the pseudo-orbit, then d(b · x, b · x0) < Kε for all b ∈ C ∩ B(0, t/2). Proof. The proof is very similar to the case of Anosov flows, with an additional observation. For simplicity of exposition, we treat the case of a single orbit segment, the case of a pseudo-orbit follows similarly by setting up a sequence of contractions (rather than a single contraction, which we describe below). By local product structure, one may take W s (x) then saturating each y ∈ W s (x) set a0,loc a0,loc with each of their local center-unstable manifolds, W cu (y) to obtain a neighborhood of x. If at ·x a0,loc 0 is sufficiently close to x, consider the following induced map on W s (x): if y ∈ W s (x), then a0,loc a0,loc at · y is also within a neighborhood of x. Define f(y) = W cu (at · y) ∩ W s (x). By the usual 0 a0,loc 0 a0,loc arguments, since the stable manifold is contracted, this is a contraction on W s (x), and has some a0,loc fixed point x ∈ W s (x). Then x satisfies atx ∈ W cu (x ). Looking at a−t on W u (x ), we 1 a0,loc 1 1 a0,loc 1 0 a0,loc 1 0 t 0 0 k may again find a fixed point to get x such that a0 · x = b · x for some b ∈ B(0, ε) ⊂ R . 0 t −1 If ε is sufficiently small and t is sufficiently large, we claim that a = a0b is still an Anosov k k element of R . Indeed, once an element of R is Anosov, so is the line passing through it (minus 0). Furthermore, the set of Anosov elements is open, so if B(a0, δ1) are all Anosov elements, so 0 are B(ta, tδ1) ⊃ B(a, tδ1) if t ≥ 1. So we may without loss of generality assume that a is Anosov. 0 0 k 0 0 Consider Fix(a ), the fixed point set of a . This set is compact, and R · x ⊂ Fix(a ). Furthermore, 0 0 k 0 since a is Anosov, Fix(a ) is a finite union of R -orbits (since if two points are both in Fix(a ) k and are sufficiently close, they must be related by an element of the central direction, ie R ). This 0 k immediately implies that x is an R -periodic orbit. Let us now obtain the estimates on d(b · x, b · x0). Notice that by the construction above, there 0 k t 0 0 0 exists a ∈ R such that d(a0, a ) < ε and a · x = x. Notice that x and x are connected by a short piece of stable manifold, and a short piece of unstable manifold, both of which have length at most ε. Similarly, a0 · x and x0 = a0 · x0 are connected by a short piece of stable manifold and short piece of unstable manifold, both of which have length at most ε. Hence, d((sa0) · x, (sa0) · 25 n 0 0 o 0 x0) < max e−λ||a ||sε, e−λ(1−s)||a ||ε , which is at most e−λs||a ||ε if s < 1/2. Notice that C0 = k 0 0 b ∈ R : d(b, sa ) < t/(2λ) for some s ∈ R+ contains C if a is sufficiently close to a0, which can be achieved by choosing ε sufficiently small. But if b ∈ C0, and d(b, sa0) < s/(2λ), then if b0 = b−ta0:

0 0 d(b · x, b · x0) ≤ eλ||b ||d(sa0 · x, sa0 · x0) < es/2 · e−λs||a ||ε < ε. k 5.3. The spectral decomposition and transitivity properties of R actions. This section provides a spectral decomposition, which we will need in the subsequent section. It may also be of k independent interest as it provides some structure to R -actions when the transitivity assumption in Theorem 1.11 is dropped. Recall that if ϕt : X → X is a continuous ﬂow on a compact metric space X, then the non- wandering set, denoted NW ({ϕt}), is the set of points x satisfying: for every ε > 0, there exist arbitrarily large t ∈ R+ such that ϕt(B(x, ε)) ∩ B(x, ε) 6= ∅. k k 0 k Theorem 5.9. Let R y X be an Anosov R action. For an Anosov element a, there exists a ∈ R arbitrarily close to a in the same Weyl chamber with the following property: there exist ﬁnitely many k k k closed R -invariant sets Λ1,..., Λn ⊂ X such that if Per(R ) denotes the set of R -periodic points:

n k [ NW ({ta}) = Per(R ) = Λi. i=1 0 Furthermore, the flow {ta } is transitive on each Λi. 0 Each Λi is called an a -basic set. Proof. First, note that from Theorem 5.8, we immediately get that NW ({ta}) = Per(Rk). To pick 0 k the element a , notice that there are at most countably many R -periodic orbits. Therefore, the set of directions whose lines are dense in every periodic orbit is generic. In particular, we may choose a0 to have this property and be in the same Weyl chamber as a. cu s Define a relation on periodic points. Say that p ∼ q if and only if Wa (p) ∩ Wa (q) 6= ∅ and cu s Wa (q) ∩ Wa (p) 6= ∅. These intersections are transverse since a is Anosov, and we claim that ∼ is an equivalence relation. Indeed, symmetry and reflexivity are trivial by construction. To k see transitivity, suppose that p ∼ q and q ∼ r, where p, q, r ∈ Per(R ). Since p ∼ q, the weak unstable manifold of p intersects the stable manifold of q. Therefore, under application of the flow, this intersection point approaches the corresponding orbit of q. Hence its weak unstable manifolds converge to the weak unstable manifold of q on arbitrarily large compact subsets. Since this weak unstable intersects the stable manifold of r, by local product structure at the intersection point, we get that p ∼ r. k Notice that by local product structure, there exists ε > 0 such that if p, q ∈ Per(R ) and d(p, q) < ε, then p ∼ q. In particular, we know that there can be only finitely many equivalence classes of ∼. Let Ei denote the union of the periodic orbits in each equivalence class, and Λi = Ei. Let cu s 0 p, q ∈ Ei. Notice that since Wa (p) ∩ Wa (q) 6= ∅, for any ε > 0, there exists p = b · p such u 0 s k 0 0 k 0 that Wa (p ) ∩ Wa (q) 6= ∅ for some b ∈ R . Since {ta · p } is dense in R · p by choice of a , 0 0 there exists arbitrarily negative s1 such that d(s1a · p , p) < ε and arbitrarily positive s2 such that 0 0 0 u 0 s 0 d(s2a · p , p ) < ε. Fix z ∈ Wa (p ) ∩ Wa (q), and let x1 = s1a · z and t1 = s2 − s1. Notice that since u 0 0 Wa (p ) is contracted by flowing backwards under a , the point z can be moved arbitrarily close to 0 0 p , and hence p. Similarly, (t1a ) · x1 can be moved arbitrarily close to q. 26 By symmetrizing the above argument, we may find x2 and t2 such that x2 is arbitrarily close to 0 0 q and (t2a ) · x2 is arbitrarily close to p. Hence for every ε > 0, there exists a closed (a , ε)-pseudo k orbit {(x1, t1), (x2, t2)}. By Theorem 5.8, there exists an R -periodic orbit shadowing it. Now, we recall the following criterion for topological transitivity of a flow on a compact metric space Λ: if U and V are arbitrary open subsets of Λ, then there exists t > 0 such that (ta0)·U ∩V 6= ∅. In our case, since Λi is the closure of a set of periodic orbits, there exists p ∈ U and q ∈ V such k that p and q are R -periodic. Therefore, by the arguments above, we may find another periodic orbit arbitrarily close to each, in particular which intersects U and V . Since by assumption, {ta0} is transitive on every periodic orbit, we conclude the criterion, and hence transitivity. 5.4. Residual Properties of Foliations. The following result is classical analog of Fubini’s theorem, and usually stated for product spaces (see, e.g., [55]). Given a foliation of a space X, one obtains that it locally looks like a product of a leaf with the transversal. This immediately gives the following: Lemma 5.10 (Kuratowski-Ulam). Let X be a smooth manifold with foliation F and Y ⊂ X be a residual subset of X. If B ⊂ X is a local transversal disc to F, then the set of points x ∈ B such that F(x) ∩ Y is residual in the leaf topology of F(x) is residual in B. We also discuss accessibility properties and their relationship to paths in foliations (recall Remark 4.5), which will appear later in the discussion.

Definition 5.11. Let Ω index a set of continuous foliations {F1,..., Fn} of a smooth manifold X. Define W Ω(x) = W F1,...,Fn to be the set of all points which can be reached using finitely many broken paths in the foliations {Fi}. The proof of the following is identical to the proof that if a flow has a dense orbit, then the set of dense orbits is residual, so we omit it. Ω Lemma 5.12. The set of points x ∈ X such that W (x) is dense is a Gδ set. If it is nonempty, it is residual. 5.5. Density of Periodic Orbits. Unlike its rank one counterpart, the following theorem is diffi- k cult to prove, even with Theorem 5.8. The difficulty lies in not knowing that if R · x is dense for k S some x ∈ X, then R · x is also dense, where R = R \ Ci and each Ci is an open cone containing a Lyapunov hyperplane. For k = 1, this is trivially true (so this complexity does not appear in rank one). k Theorem 5.13. If R y X is a transitive, totally Cartan action, then there is a dense subset of k R -periodic orbits of X. We prove Theorem 5.13 in a sequence of lemmas. The proof structure is by contradiction, so all lemmas are in the end vacuous. Let Y ⊂ X be the closure of the set of periodic orbits of X. If k Y 6= X, then U = X \ Y is an open subset of X. Since it is also R -invariant, it must be dense if it k is nonempty. Since Y is R -invariant, any dense orbit must lie in U. k Lemma 5.14. If Y 6= X, x ∈ U, and a ∈ R is a regular element, then there exists y ∈ Y such s that x ∈ Wa (y). In particular, if z = limn→∞(tna) · x, z ∈ Y . Proof. Suppose otherwise, and choose ε > 0 such that if d((ta) · x, (ta) · y) < ε for all t > 0, then cs s x ∈ Wa (y). Then if x 6∈ Wa (y) for all y ∈ Y , then f(t) = d((ta) · x, Y ) is a function which is larger than ε for infinitely many tk → ∞. By choosing a subsequence as necessary, we may arrange (tka) · x to converge, but it must converge to something outside of Y . Therefore, if xk = (tka) · x, 27 then d((t` − tk)a · xk, xk) can be arranged to be arbitrarily small, for some very large ` > k > 0. In k particular, we find a periodic orbit of R in U by Theorem 5.8, contradicting the definition of Y . k s Therefore, for every regular a ∈ R , there exists y ∈ Y such that x ∈ Wa (y). k k If x has a dense R -orbit in X, then for every z ∈ X, there exists ak ∈ R such that |ak| → ∞ k Sn Sm and ak · x = z. Recall that R \ i=1 Hi = j=1 Wj is the union of finitely many Weyl chambers, where H1,...,Hn are the Lyapunov hyperplanes of the action. If W is a Weyl chamber, let the weak interior of W be the set of sequences (an) such that an ∈ W , |an| → ∞ and d(an, ∂W ) → ∞. k We decompose the limiting points of the R orbits in the following way:

n o ωW (x) = z ∈ X : z = lim an · x, (an) is in the weak interior of W , and n→∞ n o ωH (x) = z ∈ X : z = lim (hn + bn) · x, hn ∈ H, hn → ∞, |bn| ≤ R for some R ∈ . n→∞ R

 m  n ! k [ [ Lemma 5.15. If x has a dense R -orbit, then X =  ωWj (x) ∪ ωHi (x) . Each ωWj (x) j=1 i=1 is contained in Y .

k Proof. If z ∈ X, then z = limn→∞ an · x for some an ∈ R , |an| → ∞. If an stays a bounded distance from some Lyapunov hyperplane H, then z ∈ ωH (x). Otherwise, the distance between an and every Lyapunov hyperplane tends to infinity. Since there are only finite many Weyl chambers, it must visit one of them infinitely often, call it W . Then some subsequence of (an) is in Int(W ). That is, z ∈ ωW (x). k s Finally, by Lemma 5.14, for any regular a ∈ R , x ∈ Wa (y) for some y ∈ Y . Fix some such a ∈ W . Since the elements an being applied to x do not stay within a bounded distance of the walls of W , d((an · x), (an · y)) → 0. Therefore, if z ∈ ωW (x), then z ∈ Y .

Consider a Lyapunov hyperplane H. Then H bounds ﬁnitely many Weyl chambers Wj1 ,...,Wjs . (1) (r) Let H ,...,H denote the nontrivial sets obtained as Wj ∩ H (note the the choice of j to obtain (j) H is not unique, so we simply choose one for each component). Deﬁne the sets ωH(j) in the same (j) way as ωH , where each hn must come from H rather than H (still allowing bounded variation from H(j)).

k Lemma 5.16. If x has a dense R -orbit, then ωH(j) (x) is dense and has interior in X for some Lyapunov hyperplane H and some j. Proof. Notice that we may reﬁne the union of Lemma 5.15 by taking

 m   n m  [ [ [ X =  ωWj (x) ∪  ω (j) (x) Hi j=1 i=1 j=1

R (j) Let ω (j) (x) denote the set of limits limn→∞(hn + bn) · x such that hn ∈ Hi , hn → ∞ and Hi S∞ R R |b | ≤ R. Then ω (j) (x) = ω (x), and each ω (x) is closed. That is, n H R=1 (j) (j) i Hi Hi

 m   n m ∞  [ [ [ [ R X =  ωWj (x) ∪  ω (j) (x) Hi j=1 i=1 j=1 R=1 28 We claim that one of the sets in this countable union has interior. This follows from the Baire category theorem, since each is closed. It cannot be one of the sets ωWj (x), since each is contained R in Y . Therefore, one of the sets ω (x) ⊂ ω (j) (x) has interior. (j) H Hi i k Recall that the set of dense R orbits is residual, so by Lemma 5.10, the α-leaves through a k residual set of points have a residual subset with dense R orbits. Since the set of points such that ωH(j) (x) contains an open and dense set, it is itself dense, we may assume that the point x satisﬁes: k I. x has a dense R -orbit, II. ωH(j) (x) contains an open and dense set, and (j) Let α ∈ ∆ be the coarse exponent for which H ⊂ H := ker α and α(Wj) < 0 (if α(Wj) > 0, we simply choose the chamber on the opposite side of H to deﬁne H(j)).

α Lemma 5.17. If z ∈ Int(ωH(j) (x)), then z ∈ W (y) for some y ∈ Y . s (j) Proof. Choose some a ∈ Wj. By Lemma 5.14, we know that x ∈ Wa (y0) for some y0 ∈ y. H has the same expansion and contraction properties as Wj with the exception of α. Therefore, every s (j) direction of Wa (y0) is contracted under h ∈ H with the exception of α (since by definition of (j) ωHj (x), the sequence of points along the orbit deviates a bounded amount from H ). Hence, if (hn + bn) · x → z, then (hn + bn) · y0 also converges to some y ∈ Y , and the connection from y to z α must be by W . We now choose our point x more carefully. Namely, we wish to choose it satisfying properties I. and II., but also: III. The point x ∈ W α(y) for some y ∈ Y , and is in the interior of points satisfying this property. We may choose x as in III. since by Lemma 5.17, the set of points satisfying III. is open and dense. Let Ω+ (x) denote the set of limits of x of the form lim h +b , where h ∈ H(j), α(b ) ≤ 0, H(j) n→∞ n n n n and h +b ∈ W for every n ∈ . Let ΩB (x) denote set of limits of x of the form lim h +b , n n j N H(j) n→∞ n n (j) where hn ∈ H and α(bn) ≤ 0 and bn ∈ B(0,R) for some R > 0. Since the choice of bn is bounded, by choosing a convergent subsequence and replacing bn by its limit, we may assume that the limit is of the form limn→∞ hn + b0 for some fixed b0 ∈ B(0,R). Lemma 5.18. One may choose x to satisfy the following additional properties with respect to some basic set Λ (which depends on x): IV. We have the containment ω (x) ⊂ Ω+ (x), and ΩB (x) = Ω+ (x) \ Λ. Wj H(j) H(j) H(j) V. The set Ω+ (x) is the saturation of Λ by closed segments in W α whose endpoints lie in Λ, H(j) and contains an open set. VI. The ω-limit set of {ta · x} is Λ for generic Anosov elements a such that α(a) < 0. Proof. We first show that IV. and V. are implied by assumptions I.-III. IV. is clear from construction, since if the sequence bn tends to infinity, the α leg contracts, and if the bn stay bounded, one stays α k away from Y by Lemma 5.4. By III., x ∈ W (y) for some y ∈ Y , so the R -orbit of y is contained in some basic set Λ ⊂ Y (see Theorem 5.9). Furthermore, any sequence hn + bn determining a limit in Ω+ has derivative on W α either uniformly bounded above and below (if b stays bounded), or H(j) n α converging to 0 (if α(bn) → −∞). Therefore, the closure is contained in the saturation of Λ by W . Since the bn are assumed to satisfy α(bn) ≤ 0, one stays in a fixed compact interval on one side of the intersection with some element of Y . 29 k Notice that if a ∈ R is any element such that α(a) > 0, then ωH(j) is contained in the increasing S∞ n + union a Ω . Therefore, the sets have nonempty interior, since ω (j) (x) has nonempty n=1 H(j) H interior by assumption II. This finishes the proof of V.

To see that ωWj (x) is all of Λ, notice that by Theorem 5.9, {ta} has a dense orbit on Λ (we may without loss of generality assume that a is chosen generically), so that the set of elements y ∈ Λ such that {(ta) · y : t ∈ R+} is the intersection of countably many open dense sets. Therefore, the k saturation of each open and dense subset of Y by α leaves is open and dense in R , since it is k R -invariant and open by VI. Therefore, the set of points x ∈ X such that the corresponding point y ∈ Λ has a dense orbit is residual, and hence since every other condition is residual, we may assume x belongs to this set. Fix a point x satisfying I.-VI., and let Λ be the basic set coming from VI.

α 0 0 α Lemma 5.19. Let z ∈ W (y ) for some y ∈ Λ, and set Bα(z) = W (z) ∩ Λ. Then we either have: α (1) Bα(z) = W (z) for every such z, or (2) H · x ∩ W α(x) = {x}.

Proof. Say that a closed subset S ⊂ R is syndetic if there exists R > 0 such that R \ S is a union of 0 α −1 open intervals whose length is at most R and at least 1/R. Let Bα(z) = (ψz ) (Bα(z)) ⊂ R be the 0 set Bα in normal forms coordinates. We prove that cases (1) and (2) correspond to whether Bα(x) is syndetic (where x is a point satisfying I.-VI.). 0 0 0 First, suppose that Bα(x) is syndetic. Then, applying ta to Bα(x), we get that Bα(a · x) is syndetic, with the gaps bounded by at most e−λtR, where λ is the constant given by the Anosov condition, since the dynamics in normal forms is linear. By VI., and since x ∈ W α(y) (where y is as in III.), the ω-limit set of x for the ﬂow {ta} coincides with that of y, which is all of Λ. Therefore, 0 0 0 any y ∈ Y can be accumulated by some sequence tka · x. But since Bα(a · x) = a · Bα(x), and the 0 0 0 α 0 sets can only increase in the limit, we get that Bα(y ) = R, and hence Bα(y ) = W (y ). Therefore, 0 if Bα(x) is syndetic, we are in Case (1). 0 We will now show that if Case (2) fails, we get that Bα(x) is syndetic. Indeed, if hn · x converges α α −1 α to a point x1 ∈ W (x) distinct from x, then the map f = (ψx ) ◦ lim hn ◦ ψx is a translation by 0 Lemma 5.5(b). The translation cannot be the identity since x1 6= x. Therefore, Bα contains the orbit of a translation, which is syndetic. Lemma 5.20. In case (1) of Lemma 5.19, Y = X. Proof. It suﬃces to show that Y contains an open set. Indeed, once it contains an open set, it must k k be dense since it is R -invariant and there is a dense R -orbit. Once it is dense, it must be X since Y is also closed. α By V. and case (1), Y contains an open set, since the full saturation of Λ by W has interior.

Lemma 5.21. In case (2) of Lemma 5.19, there exists points z1 6= z2 (which we may assume are α −α arbitrarily close) such that z1, z2 ∈ W (x) and h · z1 ∈ Wloc (z2) for some h ∈ H. Proof. Let x0 ∈ W α(x) be any point along the W α-segment connecting x and Y . Then by V., there 0 k 0 0 0 0 exists b ∈ R and hn ∈ H such that (hn + b ) · x → x . We claim that b 6= 0. Indeed, if b = 0, then x0 ∈ H · x. Then since we are in case (2), we would conclude that x0 = x. This is a contradiction, so b0 6= 0. Let z = (b0)−1 · x0, so that z ∈ H · x. Fix an element b ∈ H which is not contained in any other α k Lyapunov hyperplane. Then notice that since z is connected to x by a W -leaf and an R -orbit piece, d(tb · x, tb · z) stays a uniform distance from x by Lemma 5.4. By choosing x0 suﬃciently 30 close to x, we may assume that this neighborhood is one where we have local product structure for a perturbation of b to an Anosov element. Since the points stay close under tb for all positive and negative values of t, we know that the unstable and stable connections between x and z must actually be by α and −α. Setting z1 to be the endpoint of the α leg on the connection between x 0 0 and z, z2 = x and h = b , we get the result. Proof of Theorem 5.13. We must show that Y = X. This is exactly the conclusion of Lemma 5.20 in case (1) of lemma 5.19. In case (2) of Lemma 5.19, we get Lemma 5.21. However, the conclusion of the Lemma is a contradiction to local product structure. Therefore, we conclude that Y = X. We now establish several important applications of Theorem 5.13.

k k Corollary 5.22. Let R y X be a transitive, totally Cartan action. Fix a ∈ R . Then the set of n points x ∈ X such that there exists a sequence nk → +∞ such that a k · x → x is a residual subset of X.

k k Proof. Every R -periodic point is recurrent for every a ∈ R , so this is immediate from the fact that the set of recurrent points is a Gδ set. k We expect the following to hold in general for transitive Anosov R -actions. One may compare its statement with the fact that the time-t map of an Anosov flow is transitive for generic t ∈ R if the flow itself is transitive, or that, for measure-perserving transformations, the set of directions elements which are ergodic as transformations are the complement of a countable union of affine hyperplanes [58].

k Corollary 5.23. Let R y X be a transitive, totally Cartan action. Then for a generic direction a ∈ Sk−1, there exists x ∈ X such that {(ta) · x : t > 0} is dense in X.

k k−1 Proof. If p is an R -periodic orbit, let π(p) ⊂ S be the set of directions which lie in a rational subtorus. Then π(p) is a countable union of closed, nowhere dense subsets (corresponding to the S rational subtorus obtained as the closure). Let P = p π(p) be the union of such directions. Then k−1 k−1 k−1 S \ P is a dense Gδ subset of S . We claim that for any a ∈ S \ P , the forward orbit under a is dense. It is enough to show that for any open sets U, V ⊂ X, there exists t > 0 such k that (ta) · U ∩ V 6= ∅. By Theorem 5.13, for any ε > 0, there exists an ε-dense R -periodic orbit. Since U and V are open, there exists some ε > 0 such that U and V both contain ε-balls. Choose an ε/2-dense periodic orbit which must intersect U and V by construction. By assumption, the forward orbit of a is dense inside the periodic orbit, and therefore intersects both U and V . This proves the result. As indicated in Section2, the existence of a dense H-orbit is intricately related to the existence of a rank one factor, which is important in Parts I and II. We therefore wish to establish an easier way to detect whether H has a dense orbit. Fix some a ∈ H which is regular (so that β(a) 6= 0 for all β 6= ±α), and let R0 denote the set of points for which there exists sequences nk, mk → +∞ n −m such that a k · x, a k · x → x. Notice that by Lemma 5.22, R0 is residual. Then let R ⊂ X denote s u the set of points for which R0 ∩ Wa (x) and R0 ∩ Wa (x) is residual. Notice that by Lemma 5.10, R is also residual. Lemma 5.24. For every x ∈ R, W H (x) ⊂ H · x.

s nk Proof. Fix x ∈ R, and let y ∈ Wa (x) ∩ R. Then there exists nk, mk such that a · x → x and amk · y → y. Notice that since x and y are stably related, ank · y → x and amk · x → y. Therefore, s s s y ∈ H · x. Since R ∩ Wa (x) is residual in Wa (x) (by definition of R), Wa (x) ⊂ H · x. Similarly, 31 u Wa (x) ⊂ H · x. By induction, for any broken path whose break points all belong to R, the endpoint of such a path is in H · x. Since any path can be accumulated by such a path, the claim follows. Corollary 5.25. If there exists a point x such that W H (x) is dense, then there exists a point x such that H · x is dense. 5.6. Exact Hölder Metrics. Kalinin and Spatzier proved the following cocycle rigidity theorem in [42, Theorem 1.2] which is fundamental to the developments in Part II. We remark that this is not a general cocycle result but rather applies specifically to the derivative cocycle. The starting point of the argument is Lemma 5.1. One then uses a Livsic-like argument to solve the derivative cocycle along a dense H-orbit, and extend it continuously to the closure. 1,θ k Theorem 5.26. Let α be a C , totally Cartan action of R , k ≥ 2, on a compact smooth manifold M and H be a Lyapunov hyperplane for a coarse Lyapunov foliation W. If H has a dense orbit, k then there exists a Hölder continuous Riemannian metric g on T W and a functional χ : R → R k such that for any a ∈ R χ(a) ||a∗(v)|| = e ||v|| for any v ∈ Eχ. Remark 5.27. One may notice that the assumptions of [42, Theorem 1.2] are stronger than those given here. However, an inspection of the proof reveals that one does not need an invariant measure or dense orbits of one parameter subgroups: for this theorem in their paper, the conditions listed here are sufficient. In view of Theorem 2.1, Theorem 2.2 is a slightly stronger version. Remark 5.28. Theorem 5.26 can be regarded as a strengthening of Lemma 5.4 under the assumption that a Lyapunov hyperplane has a dense orbit. In particular, it implies that the Lyapunov exponent corresponding to χ for every invariant measure is equal to χ. k Remark 5.29. If R y X is a totally Cartan action, and every Lyapunov hyperplane has a dense orbit, the functionals in Definition 3.8 can be chosen to be those of Theorem 5.26. See the discussion at the start of Part II. k − 5.7. Circular orderings and geometric commutators. Fix an Anosov element a ∈ R , ∆ (a) = {β ∈ ∆ : β(a) < 0}, and Φ ⊂ ∆−(a) be a subset. We introduce a (not necessarily unique) order on 2 ∼ k the set Φ. Choose R = V ⊂ R which contains a and for which γ1|V is proportional to γ2|V if and only if γ1 is proportional to γ2 for all γ1, γ2 ∈ Φ (such choices of V are open and dense). Fix some ∗ ∗ ∼ 2 nonzero χ ∈ V such that χ(a) = 0 (χ is not necessarily a weight). Then β|V ∈ V = R for every β ∈ Φ and Φ|V = {β|V : β ∈ Φ} is contained completely on one side of the line spanned by χ.

Definition 5.30. The ordering β < γ if and only if ∠(χ, β|V ) < ∠(χ, γ|V ) is called the circular ordering of Φ (induced by χ and V ) and is a total order of Φ. If the set Φ is understood, we let |α, β| denote the set of weights γ ∈ Φ such that α ≤ γ ≤ β. While each β ∈ ∆ is only defined up to positive scalar multiple (see Remark 3.9), this is still 2 well-defined since the circular ordering on R is invariant under orientation-preserving linear maps. Figure1 exhibits such a circular ordering, after intersecting each ker χi with the generic 2-space V . The lines through 0 represent the kernels of the weights, and the arrows indicate the half space for which χi has positive evaluation. Definition 5.31. If α, β ∈ ∆, let D(α, β) ⊂ ∆ (called the α, β-cone) be the set of γ ∈ ∆ such that 2 γ = tα + sβ for some t, s ≥ 0. We may identify D(α, β) as a subset of the first quadrant of R by using the coordinates (t, s). The canonical circular ordering on D(α, β) is the counterclockwise order in the first quadrant. 32 ker χ4

ker χ3

ker χ2

kerχ 1

ker χ0

Figure 1. Circular Ordering

It is clear that the canonical circular ordering is itself a circular ordering by choosing V in the following way: ﬁrst choose a and b for which α(a) = −1, α(b) = 0, β(a) = 0 and β(b) = −1. Then perturb them to Anosov elements a0 and b0 which both contract α and β. Setting V to be the span of a0 and b0 and χ to be any functional for which χ(a0) = −1 and χ(b0) = 0 obtains the canonical circular ordering on D(α, β). Also observe that the canonical circular ordering on D(α, β) is the opposite of the canonical circular ordering on D(β, α). Let a , . . . , a ∈ k be Anosov elements, and Es = Tn Es and W s (x) = Tn W s (x). 1 n R {ai} i=1 ai {ai} i=1 ai Each W s is a Hölder foliation with Cr leaves by Lemma 3.4. Let {ai}

− ∆ ({ai}) = {γ ∈ ∆ : γ(ai) < 0 for every i} .

Lemma 5.32. Let α and β be linearly independent weights and W1,...,Wm be the Weyl chambers k of the R action such that α and β are both negative on every Wi. Then if aj ∈ Wj are regular − elements in each chamber, D(α, β) = ∆ ({ai}). Proof. That D(α, β) is contained in the right hand side is obvious. For the other, suppose that χ satisfies χ(aj) < 0 for every j = 1, . . . , m. We first show χ is a linear combination of α and β. k Suppose that χ is linearly independent of α and β. Then there exists b ∈ R such that χ(b) = 1 and α(b) = β(b) = 0. Choose any aj, and consider aj + tb. Notice that any Weyl chamber this curve passes through must be one of the Wi, since the evaluations of α and β will still be negative. But for sufficiently large t, χ(aj + tb) > 0. Hence there is a Weyl chamber such that α and β are both negative, but χ is positive. This contradicts the fact that χ is negative on each Wi, so by contradiction, χ must be linearly dependent with α and β. So we may write χ = tα + sβ. We must show that t, s ≥ 0. Indeed, assume t < 0. Since α and k β are linearly independent, it follows that there exists b ∈ R such that α(b) = 1 and β(b) = 0. Choosing any aj from the given elements, the curve aj − rb, r ∈ R satisfies β(aj − rb) = β(aj) < 0, 33 α(aj − rb) = α(aj) − r < 0 if r ≥ 0, and χ(aj − rb) = tα(aj) − tr + sβ(aj). Thus, if r is sufficiently large, α and β are negative but χ is positive. Hence there is a Weyl chamber on which α and β are negative but χ is positive. This is a contradiction, so t ≥ 0. The argument for s is completely symmetric. − Lemma 5.33. Let α, β ∈ ∆ satisfy α 6= ±β, {ai} be Anosov elements such that α, β ∈ ∆ ({ai}) − and γ1, . . . , γn be all weights of ∆ ({ai}) strictly between α and β in some circular ordering. Let − δ ∈ ∆ ({ai}) be either the next weight to α or closest to β. Then

n M TW |α,β| := TW α ⊕ TW β ⊕ TW γi and TW |α,β| ⊕ TW δ i=1 |α,β| 0 n+2 |α,β| integrate to foliations W and W , respectively. Furthermore, if ψx : R → W (x) and δ n+2 0 ϕx : R → W (x) are a continuous family of parameterizations, then the map Φx : R ×R → W (x) defined by letting Φx(u, t) = z, where y = ψx(u) and z = ϕy(t), is a homeomorphism onto its image. k Tm s Proof. Notice that we may find regular elements a1, . . . , am ∈ R such that W = i=1 Wai is a foliation such that TW |α,β| = TW . We may do the same thing when adding δ, since by assumption it is adjacent to α or β in a circular ordering and the entire collection can be placed in a stable manifold. 0 To see that the map Φx is a homeomorphism, observe first that it is onto a neighborhood of x in W since the foliation W δ is transverse to W |α,β|. We wish to show it is injective. Since δ is the boundary of a cone (either |α, δ| or |δ, β|), we may find b ∈ ker δ such that α(b), β(b), γ1(b), . . . , γn(b) < 0. Assume there exists (u, t) and (v, s) such that Φx(u, t) = Φx(v, s). Then iterating b forward, the α, γ1, . . . , γn, β-legs will all contract exponentially, but the δ leg have its length bounded away from 0 and ∞ by Lemma 5.4. By choosing a convergent subsequence, we may conclude that t = s. Then since ψx is a coordinate chart itself, we conclude u = v. Therefore, the map is a local homeomorphism near 0. That it is a global homeomorphism follows by intertwining with the hyperbolic dynamics.

Lemma 5.34. Let α, β ∈ ∆ be nonproportional weights and write D(α, β) = {α, γ1, . . . , γn, β} in the canonical circular ordering. Then given x ∈ X, x0 ∈ W α(x) and x00 ∈ W β(x0), there exists a unique y ∈ W β(x) and y0 ∈ W α(y) such that x and y0 are connected by a broken path in the W γi -foliations 0 with combinatorial pattern (γ1, . . . , γn). The length of the broken path depends continuously on x and x00. − Proof. Notice that, by Lemma 5.32, D(α, β) = ∆ ({ai}) for some choice of Anosov elements {ai}. We may iteratively apply Lemma 5.33 to obtain a chart for W |α,β|. Observe that we may begin with listing the γi-foliations listed in a circular ordering, adding them one at a time, to obtain a chart |γ ,γn| for W 1 , which is obtained by moving along each of the foliations γi, one at a time. Since each α and β bounds the collection, we may then move by β and then by α to obtain a chart for W |α,β| which first moves along the γi foliations, then the α-foliation and finally the β foliation. Choosing n+2 00 0 the element of R which maps x and x yields the desired points y and y as intermediate break points of the path. 6. Examples of Totally Cartan Actions and Classification of Affine Actions We summarize some well-known classes of Cartan actions, as well as some lesser-known examples. k There are three principle “building blocks,” (suspensions of) affine Z actions on nilmanifolds, Weyl chamber flows and Anosov flows on 3-manifolds. These actions can be combined by taking direct products, and in the homogeneous setting can also appear as fibrations (see Section 6.1.3). Theorem 1.11 shows that this is a complete characterization of Cartan actions with trivial Starkov component. 34 6.1. The Algebraic Actions. We first recall several algebraic actions, which are often called the standard actions.

k 6.1.1. Z affine actions and their suspensions. Let A1,A2,...,Ak ∈ SL(n, Z) be a collection of m1 mk commuting matrices such that A1 ...Ak 6= Id unless mi = 0 for every i. Then there is an k n n m1 m2 mk n associated action Z y T defined by m · (v + Z ) = A1 A2 ...Ak v + Z , which is an action by automorphisms. n Notice R splits as a sum of common generalized eigenspaces for each matrix Ai (where a generalized eigenspace can contain Jordan blocks, and we identify eigenvalues of the same modulus), n L` k k R = i=1 E`. There are ` different functionals on Z , which associate to m ∈ Z the modulus of m1 mk χi(a) the eigenvalue of A1 ...Ak on Ei, e . If the functionals χi are all nonvanishing, the action is Anosov, and in this case, χi are the Lyapunov functionals for every invariant measures. If each each generalized eigenspace Ei is 1-dimensional and there are no positively proportional Lyapunov functionals, the action is totally Cartan. Notice that the totally Cartan condition implies that there are no Jordan blocks of the action. Analogous actions can be constructed when replacing n T by a nilmanifold, see [61]. We will restrict our discussion to tori for simplicity. k We have already described a dynamical suspension construction for Z actions in Section 3.5. Here, we show that in the case of Cartan actions by automorphisms, this is also realized as a k n homogeneous flow on a solvable group S = R n R . We define the semidirect product structure k k of S by fitting the Z subgroup of SL(n, Z) into an R subgroup. Since the eigenvalues of each Ai are all positive real numbers, each Ai fits into a one-parameter subgroup, Ai = exp(tXi) with Xi ∈ sl(n, R). Furthermore, since [Ai,Aj] = e, [Xi,Xj] = 0. Thus, there exists a homomorphism k k f : R → SL(n, R) such that f(ei) = Ai for every i (so in particular, f(Z ) ⊂ SL(n, Z). k n We are ready to define the semidirect product structure of S. Let (ai, xi) ∈ R × R for i = 1, 2, and define

−1 (a1, x1) · (a2, x2) = (a1 + a2, f(a2) x1 + x2) k n k n k Then Γ = Z × Z is a cocompact subgroup of R × R and R is an abelian subgroup. The k n translation action by R is a Cartan action, since the common eigenspaces in R are the coarse n k Lyapunov subspaces. Furthermore, the stabilizer of the subgroup T ⊂ S/Γ is exactly Z , and by n k construction, if v ∈ R and a ∈ Z :

a · v = f(a)v · a ∼ f(a)v

k n Therefore, the translation action on S/Γ is the suspension of the Z action on T .

6.1.2. Weyl Chamber Flows. Let g be a semisimple Lie algebra and a ⊂ g be an R-split Cartan subalgebra, which unique up to automorphism of g. Cartan subalgebras are characterized by the following: for every X ∈ a, adX : g → g diagonalizable over R, and is the maximal abelian subalgebra satisfying this property. A semisimple algebra g is called (R-)split if the centralizer of a (ie, the common zero eigenspace of adX for X ∈ a) is a itself. The semisimple split Lie algebras are well-classified, the most classical example being g = sl(d, R), P ∼ d−1 with Cartan subalgebra a = {diag(t1, t2, . . . , td): ti = 0} = R , which we will address directly now. Other examples include g = so(m, n) with |m − n| ≤ 1 and g = sp(2n, R), as well as split forms of the exotic algebras. In what follows, SL(d, R) may be replaced with a split Lie group G, with its corresponding objects. 35 Let Γ ⊂ SL(d, R) be a cocompact lattice, and define the Weyl chamber flow on SL(d, R)/Γ to be t1 td P ∼ d−1 the translation action of A = diag(e , . . . , e ): ti = 0 = R . Notice that if Y ∈ sl(d, R) = TeSL(d, R), and a = exp(X) ∈ A with X ∈ a, then:

da(Y ) = Ada(Y ) = exp(adX )Y. α(X) Therefore, if Y is an eigenvector of adX with eigenvalue α(X), da(Y ) = e Y . The eigenvectors th are exactly the elementary matrices Yij, with all entries equal to 0 except for the (i, j) entry, which is 1. For a general split group, it is classical that the eigenspaces are 1-dimensional. By direct computation, if X = diag(t1, . . . , td), then adX (Y ) = XY − YX = (ti − tj)Y . Therefore, the eigenvalue functionals α are exactly α(X) = ti − tj. These functionals α are called the roots of g, and are the weights of the Cartan action as considered above. For a general semisimple Lie group, the translation action of an R-split Cartan subgroup is not Cartan, or even Anosov. However, the centralizer of a is always isomorphic to a ⊕ m, where m ⊂ g is the Lie algebra of some compact Lie subgroup M ⊂ G. In this case, exp(a) descends to a left translation action on the double quotient space M\G/Γ, and this action is totally Anosov.

6.1.3. Twisted Weyl Chamber Flows. This example is a combination of the previous two examples. Let G be an R-split semisimple Lie group, and ρ : G → SL(N, R) be a representation of G, which has an induced representation ρ¯ : g → sl(N, R). Then ρ¯ has a weight α : a → R for each common N eigenspace E ⊂ R for the transformations {ρ(X): X ∈ a}, which assigns to X the eigenvalue of ρ(X) on E. We call ρ a Cartan representation if: (1) (No zero weights) 0 is not a weight of ρ¯ (2) (Non-resonant) no weight α of ρ¯ is proportional to a root of g1 (3) (One-dimensional) the eigenspaces Eα for each weight α are one-dimensional. Given an R-split semisimple group with Cartan representation ρ, we may deﬁne a semidirect N N product group Gρ = G n R which is topologically given by G × R , with multiplication deﬁned by:

−1 (g1, v1) · (g2, v2) = (g1g2, ρ(g2) v1 + v2) We now assume that Γ is a (cocompact) lattice in G such that ρ(Γ) ⊂ SL(N, Z) (this severely N restricts the possible classes of ρ and Γ one may take). Then Γρ = Γ n Z ⊂ Gρ is a (cocompact) lattice in Gρ, and the translation action of the Cartan subgroup A ⊂ G on Gρ/Γρ is the twisted Weyl chamber flow. The weights of the action are exactly the roots of g and weights of dρ¯, which by the non-resonance condition implies that the coarse Lyapunov distributions are all one-dimensional. N When R is replaced by a nilpotent Lie group, one may replace the toral fibers with certain nilmanifold fibers. See [61].

k 6.2. Classification of Affine Cartan Actions. Here we consider affine actions of R by left k l translations or more generally R × Z by left translations and automorphisms on homogeneous spaces G/Γ where G is a connected Lie group, and Γ ⊂ G a uniform lattice. Unlike the rest of the paper, we do not require k + l ≥ 2 to obtain classification under the homogeneity assumption. Below we describe the structure of the homogeneous space G/Γ and the action α in terms of the Levi decomposition.

1In fact, this is implied by (1). 36 Theorem 6.1. Suppose that G is a connected Lie group, and Γ ⊂ G a cocompact lattice. Suppose k l R × Z y G/Γ is an affine Cartan action. Let S and N denote the solvradical and nilradical of G, ∼ respectively, and G = L n S be a Levi decomposition of G for some semisimple group L. Then (1) If σ : G → L is the canonical projection, then σ(L) is a lattice in L. (2) S ∩ Γ is a lattice in S. k (3) The restriction of the Cartan action to R covers a Weyl chamber flow on L/σ(Γ), and the l action of Z factors through a finite group action by automorphisms of L which preserve σ(Γ). (4) N ∩ Γ is a lattice in N, and if χ is a coarse Lyapunov exponent which does not restrict to a root of L, then Eχ ⊂ Lie(N). k ` Proof. Note that the suspension of an affine R × Z action is again affine. Thus it suffices to argue k the case of an affine action by R . (1), (2) and the first part of (4): These follow from Corollary 8.28 of [63], provided L has no compact factors. To see that there are no such factors, observe that since the Cartan action is affine, k l a · (gΓ) = (f(a)ϕa(g)Γ), where f : R × Z → G is some homomorphism and ϕa is an automorphism of G preserving Γ. This implies that da = Ad(f(a))◦dϕa, which is an automorphism of g = Lie(G). k The sum of the eigenspaces of modulus 1 for this automorphism must be Lie(f(R )), since the action is Cartan. Every automorphism of a compact semisimple Lie group has only eigenvalues of modulus 1. But semisimple groups are not abelian, so the Lie algebra cannot only consist of the k R -actions. Therefore, we may apply the result. (3): We claim that a · Sx = S(a · x). Since S is a characteristic subgroup, it is invariant under k l the automorphism ϕa as well as conjugation by an element of G. Since the action of R × Z is a composition of such transformations, we get the intertwining property. In particular, the Cartan action descends to an action on L/σ(Γ). For any semisimple group, the identity component of k k l Aut(L) is the inner automorphism group. Therefore, the R component of R × Z must act by translations, since there are no small g ∈ L such that gσ(Γ)g−1 = σ(Γ). Furthermore, the induced action must have some continuous part if L 6= {e}. Indeed, since the outer automorphism group l l of G is finite, if the action were by Z , some finite index subgroup of the Z action would be a translation action. But the elements of G may be written as z · a · u, where z is in the center of G, a is in some Cartan subgroup and u is a nilpotent element commuting with a. In particular, the one-parameter subgroup generating a would be a 1-eigenspace of the action which is not contained in the orbit, violating the Cartan condition. Therefore, the action must contain at least one one- parameter subgroup which acts by translations. By the Cartan assumption, da is diagonalizable for every a in this subgroup, so the translations must be by semisimple elements. Furthermore, the action must contain all elements that commute with a, and therefore an R-split Cartan subgroup. k l l That is, the action of the continuous part of R × Z is a Weyl chamber flow and the Z factors through a finite group action by automorphisms fixing the corresponding Cartan subgroup. To see the second part of (4), we note that the arguments of Proposition 3.13 in [26] can be applied to show that if Eχ ⊂ Lie(S), then Eχ ⊕ Lie(N) is a nilpotent ideal of Lie(S). This is sufficient for our claim, since the nilradical is exactly the elements of the solvradical which are ad-nilpotent. The setting there is that of an affine Anosov diffeomorphism, rather than an action which may have a subalgebra inside the group generating the action. This leads to some exotic examples (see Section 6.3.1). 6.3. Exotic examples of totally Cartan actions. In this section, we review examples of Cartan actions which exhibit unusual behavior. Many of these examples exhibit features which explain the more technical constructions and assumptions needed in Theorem 1.11. 37 k l 6.3.1. Homogeneous actions with Starkov component. The main examples of Cartan R ×Z actions on solvable groups S are Cartan actions by automorphisms and suspensions of such actions. Indeed, (4) of Theorem 6.1 implies that the subgroup transverse to n, the nilradical of S, is a piece of k the R -orbit. However, the orbit may also intersect n. This may happen in a trivial way, by k l taking any R × Z action and its direct product with a transitive translation action on a torus. k There are other interesting examples as well, and in each such example, the component of the R - action intersecting n is always part of the Starkov component. For instance, one may construct 2 2 2 an R action in the following way: let A : T → T by a hyperbolic toral automorphism. Then A also induces a diffeomorphism of the nilmanifold H/Λ, where H is the Heisenberg group H = 1 x z    0 1 y : x, y, z ∈ R and Λ are the integer points. The automorphism is given by (x, y, z) 7→  0 0 1  (A(x, y), z). While the automorphism is not Cartan, letting one copy of R act by translation along 2 the center and the other by a suspension (as in Section 6.1.1) gives an R Cartan action, since the k action will still be normally hyperbolic with respect to the R orbits for a dense set of elements. This action has a rank one factor, but similar examples can be constructed without rank one factors d 2 by taking a torus extension of T , d ≥ 4, rather than T . In an analogous way, assume that a Cartan representation of a split Lie group G preserves a N N symplectic form ω on R . Then one may define a Heisenberg extension 1 → R → H → R → 1 N N of R by letting H = R × R topologically, with multiplication (t1, v1) · (t2, v2) = (t1 + t2 + ¯ ω(v1, v2), v1 + v2). Then let Gρ = G nρ H, where ρ(g)(t, v) = (t, ρ(g)v). Since ρ preserves the symplectic form ω, the extension of ρ to H acts by automorphisms. Then by construction, the Cartan subgroup of G commutes with the center of H, and the action of A × R is a Cartan action on any cocompact quotient of G¯ρ. What makes these examples special is that they have nontrivial Starkov component, which is not a direct product with a torus action. The examples described here are extensions of totally Cartan actions by isometric actions, as described in Theorem 1.11. In the homogeneous case, the Starkov component is exactly the directions generated by any center of a Heisenberg group appearing in Proposition 12.11. Furthermore, in relation to Remark 13.9, one may construct ρ in a way to produce other interesting behavior. Indeed, fix any Cartan representation ρ0 of G for which every weight α of the ∗ representation has cα not a weight for any c 6= 1. Then ρ0 ⊕ρ0, the direct sum of this representation ∗ with its dual, is a Cartan representation of G. Furthermore, the weights of ρ0 are the opposites of the weights of ρ0, and there is an invariant symplectic form which pairs a weight with its opposite. We may construct the corresponding Heisenberg extension of the twisted Weyl chamber flow, and in this case, if ∆ρ0 is the set of weights for ρ0, this is an ideal in the sense of Definition 13.1. However, one easily sees that while every χ ∈ ∆ρ0 generates the Heisenberg group with its corresponding −χ ∈ ∆ ∗ , −χ is not in the ideal. ρ0

6.3.2. Starkov component arising from mixing semisimple and solvable structures. The following construction is due to Starkov, and first appeared in [47]. Consider the geodesic flow of a compact surface, realized as a homogeneous flow on SL(2, R)/Γ, and the suspension of a hyperbolic toral 2 2 automorphism A : T → T which lies in a one-parameter subgroup of matrices in SL(2, R), t 7→ At. We first repeat the construction above: take the central extension of the suspension flow to obtain a flow on a homogeneous space of a group H˜ = R n H, where H is the Heisenberg group, t and where R acts on H by a hyperbolic map (t 7→ A ) on a subspace transverse to Z(H), and trivially on Z(H) = Z(H˜ ). We assume the lattice takes the form Λ = Z n H(Z), and notice that 38 ˜ Z0 := Z(H) ∩ H(Z) is isomorphic to Z. Since Γ ⊂ SL(2, R) is cocompact, it is the extension of ∼ some surface group. Therefore, there are many homomorphisms from Γ to Z(H) = R, fix such ∼ a nontrivial homomorphism f such that the image of f is not commensurable with Z0 = Z. We construct a quotient of the space SL(2, R) × H˜ by considering the subgroup which is the image of Γ × Λ under the map (γ, λ) 7→ (γ, λf(γ)). Call this subgroup Γ˜. Notice that Γ˜ is still discrete, since if (γn, λnf(γn)) → (e, e), then γn is eventually e since Γ is a lattice in SL(2, R). Hence λn is eventually e as well. Thus, there is a well-defined, homogeneous 3 R action on X = (SL(2, R) × H˜ )/Γ˜ which flows along the diagonal in SL(2, R), the R direction of H˜ = R n H and Z(H) as the three generators of the action. Notice also that the action has a rank one factor, since the projection onto the first coordinate is well-defined. However, the action on H/˜ Λ is not a factor, since the projection of the lattice is dense in the center of H (by the non-commensurability assumption on f). This example highlights the necessity of the assumptions of Theorems 1.10 and 1.11, since it has a Starkov component, namely Z(H). Indeed, the action of Z(H) factors through a circle action on X, and after quotienting by this circle action, one arrives at the direct product of the flow on SL(2, R)/Γ and the suspension of A. Remark 6.2. One can make this example non-homogeneous by using the unit tangent bundle to a surface of negative, non-constant curvature, rather than SL(2, R) in this construction. This highlights further the need for quotienting by the Starkov component in the construction (since if the flow is homogeneous, it is already one of the cases of Theorem 1.11). 6.3.3. Suspensions of Diagonal Actions. Consider a collection of totally Cartan actions by linear 2 automorphisms Ai : Z y T , i = 1, . . . , k, which may or may not be related in any way. Then let ` k k 2 ≤ ` < k, and choose any homomorphism f : Z → Z such that if πi : Z → Z is the projection th onto the i coordinate, then πi ◦ f and πj ◦ f are both nonzero and nonproportional. Such choices are always possible for ` ≥ 2. ` 2k Then construct the action of Z on T by:

f(a)1 f(a)k a · (x1, . . . , xk) = (A1 (x1),...,Ak (xk)) ` ` Notice that we may suspend the Z action to an R action. Furthermore, the Lyapunov exponents of the action are exactly πi ◦ f, so by construction the action is totally Cartan. The action has ` trivial Starkov component, but is not virtually self-centralizing. In fact, the centralizer of the R ` k−` action is exactly R × Z , since one may act by any of the automorphisms Ai in its corresponding torus. In this action, we see the necessity for the technicalities of Theorem 1.11: in Theorem 1.10, we ` obtain the action as a finite extension of a product of standard actions. The R action we have k k constructed cannot be described this way. If one suspends the full Z -action, one obtains an R - action which is the direct product of k suspension flows, as described by Theorem 1.10 when the action is assumed to be virtually self-centralizing. The suspension space for this action is locally k 2k 2 given by R ×T = (s1, . . . , sk; x1, . . . , xk): si ∈ R, xi ∈ T , and the flow is given by translations ` ` in the si coordinates. The embedding of R is rational and therefore f(R ) is the kernel of some k k−` ` rational homomorphism g : R → R . Thus we see that while the R action is not itself a product action, it embeds in a product action. It is exactly the restriction of a product action to some rational subspace, which is precisely the content of Theorem 1.11. 6.3.4. Non-Totally Anosov Actions. The classical Katok-Spatzier Conjecture 1.5 is stated for Anosov, and does not require them to be totally Anosov actions. In most results in the theory (including 39 the results of this paper), the totally Anosov condition assumed. In fact, the only results in which Anosov and not totally Anosov is assumed are [64, 66]. For rank one actions, it is easy to see that these conditions are equivalent. Here we show that for higher-rank actions, Anosov does not imply totally Anosov. Note that this is not a counterexample to the conjecture, since the example we construct has a rank one factor. We will produce an example below, assume it is totally Anosov, and get a contradiction. Fix some cocompact Γ ⊂ SL(2, R), let X = SL(2, R)/Γ and Y = X × X. Let vi, i = 1, 2 be 1 0 the vector fields generated by ∈ sl(2, ) in the first and second factor of Y , respectively. 0 −1 R Let ϕ : X → R be a function which has an average of 0, and not cohomologous to a constant. Let w1(x1, x2) = v1 + ϕ(x1)v2 and w2(x1, x2) = v2. Notice that:

[w1, w2] = [v1 + ϕ(x1)v2, v2] = −[v2, ϕ(x1)v2] = Dv2 ϕ(x1)v2 = 0 2 Therefore w1 and w2 generate an R action. Let a = w1 + sw2, where s is such that ϕ(y) ≤ s + 1 for every y ∈ Y . Now, notice that the derivative of a on Y still preserves the same unstable bundle for v1 + v2, and uniformly expands it. Similarly, it contracts the corresponding stable bundle, so a 2 is an Anosov element of the R action. Now, choose s, t ∈ R such that b = tw1 + sw2 = tv1 + (tϕ(x1) + s)v2 is an element such that tϕ(x1) + s has positive integral on some periodic point x1 of v1 on X and negative integral on another periodic point x2. That is, y1 = (x1, x1) is periodic for the w1, w2-action and the integral of x 7→ s + tϕ(x) over the base periodic orbit is < −ε, and x2 and y2, respectively so that the integral 0 1 of s + tϕ over the base is > ε. Let u = be the vector field generating the unstable bundle in 0 0 the second (fiber) factor. Then at y1, u is contracted by, and invariant under b, and any sufficiently 2 small perturbation of b within R must contract u. Similarly, at y2, u is expanded by, and invariant 2 under b, so any sufficiently small perturbation of b within R must expand u. Therefore, u cannot be contained in a coarse Lyapunov subspace, since the coarse subspaces are uniformly expanded or contracted for a dense set of elements, by totally Anosov. But since it is invariant under the w1, w2-action, it must be either part of the action or a coarse Lyapunov subspace if the action were totally Anosov. This is a contradiction, so that action is not totally Anosov.

6.3.5. Actions with orbifold, but not manifold, rank one factors. Throughout, we have assumed α k that each foliations W is orientable. Without this assumption, there are examples of R actions in which the factor X/W H is an orbifold, but not a manifold (for a definition of X/W H , see Section 9). We outline one construction here. Let S1 and S2 be hyperbolic surfaces, each having an order 2 isometry. For S1, we assume there is an isometry f¯1 : S1 → S1 which is a reflection across a systole separating two punctured tori. Notice that Fix(f¯1) is exactly the systole. One can modify this construction to be on any surface as long as Fix(f¯1) is a union of closed geodesics. Now let f¯2 be any involutive isometry of a surface S2 which does not have fixed points. One such example can be obtained by gluing an even number of twice-punctured tori together in a cycle, then permuting them in the circle exactly halfway around. 1 Now, let Yi = T Si be the corresponding unit tangent bundles, and fi : Yi → Yi be the maps induced by the isometries. Notice that Fix(f1) is still a union of circles (exactly the orbits of the geodesic flow tangent to the systole), and that while f¯1 may reverse orientation, f1 always preserves it (since if f¯1 reverses the orientation on S1, it reverses the orientation on the circle bundle as well). Finally, if Z0 = Y1 × Y2 and f(y1, y2) = (f1(y1), f2(y2)), then f does not have fixed points 2 (1) (2) (1) (2) and commutes with the R action (t, s) · (y1, y2) = (gt (y1), gs (y2)), where gt and gs are 40 2 the geodesic flows on Y1 and Y2, respectively. Therefore, there is a totally Cartan R action on Z = Z0/(z ∼ f(z)). 2 The Lyapunov foliations of the R actions are exactly the horocyclic foliations in Y1 and Y2, respectively. Therefore, if α is the positive exponent for the flow on Y1 and β is the positive exponent H for the flow on Y2, ∆ = {α, −α, β, −β}. Notice that W (y1, y2) lifts to {y1}×Y2∪{f1(y1)}×Y2 ⊂ Z0 H H (for a discussion on W , see Sections . Therefore, in this example Z/W = Y1/(y ∼ f1(y)), which is not a manifold since f1 has fixed points.

Part I. Rank One Factors and Transitivity of Hyperplane Actions 7. Basic properties of Lyapunov central manifolds Fix a Lyapunov half-space α with Lyapunov hyperplane H. In this section, we build a model for k a non-Kronecker rank one factor of R y X assuming that the H action is not transitive. The plan to construct the factor is as follows: the non-existence of dense H-orbits provides the existence of H-periodic orbits. We then show that the ﬁxed point sets for the elements of H ﬁxing the periodic point are smooth manifolds. Quotients of these manifolds will provide candidates for the rank one factor.

7.1. Existence of H-periodic orbits. In this section, we describe certain features of H-orbits S when there does not exist a dense H-orbit. Call a subset S ⊂ X ε-dense if x∈S B(x, ε) = X. We say that a point p is periodic for a group action H y X if the orbit H· p is compact, and call StabH(p) ⊂ H the periods of p. We ﬁrst prove the following very basic topological lemma:

Lemma 7.1. Let B y X be a group action of a group B by homeomorphisms of a compact metric space X. Assume that for every ε > 0, the set {x ∈ X : B · x is ε-dense} is dense in X. Then there exists a dense B-orbit.

Proof. Pick a countable dense subset {xn : n ∈ N} ⊂ X. Then deﬁne

Yn = {y ∈ X : for all 1 ≤ ` ≤ n, there exists b ∈ B such that d(b · y, x`) < 1/n} .

By deﬁnition, any 1/n-dense orbit will be contained in Yn, so Yn is dense by assumption. It is T∞ also open by construction. Therefore, n=1 Yn is residual, and therefore nonempty by the Baire Category Theorem. It is easy to see that any element of this intersection has a dense orbit. k Corollary 7.2. Let R y X be a totally Cartan action. If there does not exist a dense H-orbit, k then there exists ε > 0 such that every ε-dense, R -periodic point is H-periodic. Furthermore, the set of H-periodic orbits are dense.

k Proof. Assume otherwise. Then for every ε > 0, there exists an R -periodic point p whose orbit is ε-dense, but not H-periodic, since the periodic orbits are dense by Theorem 5.13. The H-orbit is k dense inside the R -orbit, since H will correspond to an irrational codimension 1 hyperplane. Since 41 this holds for every ε, the union of such orbits is dense. Therefore, the set of ε-dense H-orbits are also dense. Hence, by Lemma 7.1, there exists a dense H-orbit, contradicting our assumption. To see that the set of H-periodic orbits are dense, notice that once a point is H-periodic, so is k k every point on its R -orbit. Since for every ε > 0, there exists an H-periodic orbit whose R -orbit is ε-dense, the set of H-periodic orbits are dense. Finally, we shall later see that H factors through a torus action, so we recall a criterion for the quotient by a compact group action to be a manifold. 7.2. Fixed point sets as manifolds. Recall that, in a weak sense, H = “ker α,” even though α may itself not correspond to a Lyapunov exponent. That is, H is the boundary of the half-space determined by α(a) < 0 which contracts a foliation W α. In general, −α may not be an element of ∆, if it is not, we omit it from the legs used in constructing paths throughout. If x ∈ X, let M α(x) be the set of points which can be reached from x by a broken path in the α k and −α (if −α ∈ ∆) coarse Lyapunov foliations and R -orbits. k r Proposition 7.3. Let R y X be a C totally Cartan action with r = 2 or r = ∞, and let a ∈ H not belong to any other Lyapunov hyperplane. Then if Fix(a) is nonempty, it is a Cr submanifold of X, which is (k + 2)- or (k + 1)-dimensional, depending on whether −α ∈ ∆. Furthermore, Fn α Fix(a) = i=1 M (yi) for some ﬁnite collection {yi} ⊂ X. Proof. Suppose that y ∈ Fix(a). If −α 6∈ ∆, we let W −α(y) = {y} and E−α = {0} be the 0- dimensional foliation into points and trivial bundle, respectively. We claim that W α(y),W −α(y) ⊂ α −α k Fix(a). Indeed, W and W are each contracting foliations for some elements b, −b ∈ R . There- ± ±α fore, there exist normal forms coordinates φy : R → W which intertwine the dynamics with linear dynamics (see Section 3.3). Since a ∈ H, ||da|W ±α (p)|| = 1, since otherwise, either a or −a exponentially contracts W ±α. Therefore, since the dynamics is intertwined with linear dynamics and has derivative equal to 1, it must be the identity. That is, W ±α(y) ⊂ Fix(a). Therefore, saturating k α p with α, −α and R orbits preserves the property of being ﬁxed by a, and M (y) ⊂ Fix(a). u s Use the exponential map and coordinates for TpM subordinate to the splitting TpM = Ea ⊕Ea ⊕ 0 0 k α −α Ea (where Ea = T R ⊕ E ⊕ E ) to conjugate the dynamics of a to a map of the form

F (x, y, z) = (L1x, L2y, z) + (f1(x, y, z), f2(x, y, z), f3(x, y, z)), −1 where L1 is a linear transformation such that L1 < 1, L2 is a linear transformation such that dim X dim E∗ ||L2|| < 1 and each fi : R → R a (where ∗ = s, u, 0 appropriately) satisfy dfi(0) = 0 and fi(0) = 0. Notice that if a point (x, y, z) is fixed by F , then (L1 − Id)x = −f1(x, y, z), (L2 − Id)y = −f2(x, y, z) and f3(x, y, z) = 0. Therefore, if −1 −1 g(x, y, z) = ((Id − L1) f1(x, y, z) − x, (Id − L2) f2(x, y, z) − y), then the fixed point set lies inside g−1(0). −Id 0 0 Near 0, g−1(0) is a submanifold by the submersion theorem, since g0(0) = 0 −Id 0 0 in block form. Furthermore, its dimension is exactly dim Ea = k + 2. We claim that near 0, g−1(0) = Fix(a). That Fix(a) ⊂ g−1(0) is clear from the first part of the proof, so we must show ±α k the opposite. We know that W -leaves and R -orbits lie inside Fix(a). Therefore, we can construct k −1 a locally injective map from R × R × R → Fix(a) ⊂ g (0). By invariance of domain, the map is a local homeomorphism, and Fix(a) = g−1(0) locally. Since this argument can be done at every point of Fix(a), we get that M α(y) is a Cr submanifold of X. Since there is uniform hyperbolicity transversally to each M α(y), the connected components of Fix(a) are of the form M α(y) and they are isolated, so there are finitely many. 42 Lemma 7.4. If α ∈ ∆, but −α 6∈ ∆, then H has a dense orbit. Proof. Assume otherwise. Then by Corollary 7.2, there is an H-periodic point, p. Since the set of elements of H fixing p must be a lattice in H, we may pick some a ∈ H not contained in any other Lyapunov hyperplane for which a · p = p. By Proposition 7.3, the connected component of α k Fix(a) containing p is a (k + 1)-dimensional manifold M (p) which is saturated by R -orbits. Then pick any one-parameter subgroup which contracts α. This flow must preserve the compact manifold α k M (p), contracts α, and is isometric on R . We claim that such flows do not exist on compact manifolds. Recall that p is periodic, and hence α α k its orbit is compact. Each x ∈ M (p) is connected to the orbit of p by a W -leaf, since the R -orbits α k α and W -leaves are transverse and complementary, and the R -action preserves the W -foliation. α α Let δ : M (p) → R be defined by letting δ(x) be the shortest distance along the W -leaf connecting k x to the R orbit of p. Since such a connection exists, varies continuously, and the intersections are discrete in the leaf topology of W α by transversality, δ is a continuous function. Furthermore, if δ(x) 6= 0, δ((na) · x) < δ(x) for sufficiently large n whenever α(a) < 0. Thus, the maximum of δ cannot be positive, since applying −na will yield a larger value. Hence δ ≡ 0, and M α(p) consists k of a single R -orbit. This is a contradiction. Remark 7.5. Lemma 7.4 is reminiscent of arguments used to study totally nonsymplectic (TNS) actions. The TNS condition is formulated using Lyapunov exponents with respect to an invariant measure, asking them to never be negatively proportional. One can conclude from this that the corresponding Weyl chamber wall is ergodic. The proof is more complicated and uses Pinsker partitions (this condition and argument is now widely used and first appeared in [48]). Our proof is a spiritual sibling in the topological category. From now on, we assume that ±α ∈ ∆, and that H does not have a dense orbit. k r Proposition 7.6. Let R y X be a C , totally Cartan action, with r = 2 or r = ∞. Suppose that k α p is both H- and R -periodic. Then the H action on M (p) factors through a free torus action. If Y denotes the quotient of M α(p) by H-orbits, then Y has a canonical Cr manifold structure, and the projection from M α(p) to Y is Cr.

Proof. If p is H-periodic, then StabH (p) is a lattice in H. One may choose a generating set for StabH (p) which consists only of elements which are not contained in any other Lyapunov hyperplane. α By Proposition 7.3, M (p) = Fix(a) for any such generator, so StabH (p) ⊂ StabH (q) for all α q ∈ M (p). Since StabH (q) contains a lattice, it is itself a lattice. Therefore, a symmetric argument ∼ k−1 shows that StabH (q) ⊂ StabH (p). That is, the H-action factors through H/ StabH (p) = T as a free action. That the quotient space of a free, Cr, compact group action is a Cr-manifold is classical, see e.g., [50, Theorem 21.10]. 8. W H -Holonomy Returns to Lyapunov central manifolds k We continue working with a transitive, totally Cartan action R y X, and assume that a fixed coarse Lyapunov hyperplane H = ker α does not have a dense orbit. Choose a line L transverse to k k α α H ⊂ R , and an H- and R -periodic point p ∈ X. Notice that L preserves M (p), where M (p) is as in Proposition 7.3. Let Y be as defined in Proposition 7.6. The following set will be crucial to our construction of a rank one factor: Definition 8.1. If x ∈ X, let W H (x) be the set of points y such that there exists a path ρ along the W β-foliations for β 6= ±α, and H-orbit foliations which ends at y. That is, ρ is a sequence β x = x0, x1, . . . , xn = y such that xi ∈ W i (xi−1) or xi ∈ H · xi−1 for every i = 1, . . . , n (recall 43 Remark 4.5). Let e(ρ) = y denote the endpoint of ρ and let `(ρ) be the sum of the lengths of the legs. 8.1. Local Integrability of M α. Lemma 8.2. The action of L descends to Y as an Anosov flow. Proof. That the L-action descends to Y follows from the fact that L-commutes with H, and that Y is the factor of M α(p) by H-orbits. The Anosov property is also clear: Y is a 3-dimensional manifolds with complementary foliations given by L-orbits, W α and W −α. Since α and −α are expanded and contracted, respectively, by elements of L, the flow is Anosov.

Let at denote the Anosov flow on Y as in Lemma 8.2. Fix some point y¯0 ∈ Y , let y0 be any lift of H H y¯0, and consider W (y0). Let ρ denote a W -path based at y0, and η denote a path in the α,−α k and R -orbit foliations (with e(η) and `(η) denoting its endpoint and length, respectively). We assume that we have passed to a finite cover on which the foliations W β are orientable for k every β ∈ ∆. Fix x ∈ X, and construct a function ψx : R × R × R → X in the following way: given k α t, s ∈ R and u ∈ R , let x1 denote the point at (signed) distance t from x along W , x2 denote −α the point at (signed) distance s from x1 along W and ψx(t, s, u) = u · x2. This corresponds to the evaluation map of the path group along a fixed combinatorial pattern (see Remark 4.5). The following lemma can be thought of as proving M α to be “locally topologically integrable.” That is, local C0 Euclidean structures can be constructed using ψ. Lemma 8.3. There exists L > 0 such that for every x ∈ X, if y ∈ M α(x) is reached from x by a path k η in the α, −α, R -foliations with `(η) < L (with an arbitrary number of legs), then y = ψ(t, s, u) k+2 for some (t, s, u) ∈ R with |t| + |s| + ||u|| ≤ C`(η) for some constant C. Proof. Fix some a ∈ H which does not belong to any other Lyapunov hyperplane, and let A be as in Lemma 5.4. Let L be less than the injectivity radius of X divided by A and be small enough such that if two points on an unstable or stable leaf have distance δ < 2L · A in the metric on X, 9 11 α −α their distance on the manifold is between 10 δ and 10 δ. Let η be an arbitrary path in the W , W k and R -orbit foliations with `(η) < L. Note that there may be an arbitrary number of switches between each of the foliations. Put the stable and unstable coarse exponents for a in a circular order, {β1, . . . , βr} and {γ1, . . . , γs}. Since each of the coarse Lyapunov foliations are transverse k and complementary when put together with the R -orbit foliations, we may choose a path beginning k at x and ending at e(ρ) which first moves along α, then −α, then R , then β1 through βr and finally γ1 through γs (see, eg, [68, Lemma 3.2] and compare with Lemma 12.5) (shrinking L as needed). Assume, for a contradiction, that one of the legs in the βi foliations or γi foliations is nontrivial. Suppose without loss of generality, that one of the {γi} is nontrivial. Since the γi are listed in a circular ordering, moving along each in sequence parameterizes the expanding foliation by iteratively applying Lemma 5.33. In particular, once one of them is nontrivial, the points at the start and end of the γi-legs will be distinct points on the same unstable leaf. Then iterating a the unstable leaf will expand, and the stable leaf will contract. Since a ∈ H, the endpoint after moving along the k α-legs and R -orbit will stay within the ball of radius L · A. But a pair of distinct points in the unstable manifold of a will eventually grow to a distance larger than L · A, a contradiction. This k implies that only the α and −α-legs, and R -orbits have nontrivial contribution. That is, that y is in the image of ψx. Finally, we show that (t, s, u) is unique and depends in a Lipschitz way on `(η). Perturb a to a regular element which now expands α and contracts −α. Observe that any pair of points can be locally reached uniquely by first moving along the the unstable foliation, then the stable 44 k foliation, then the R -orbit foliation. This immediately implies uniqueness. The lengths of each leg are controlled by the distance on the manifold, so we get the bound on the length of the path as described. α α We introduce the following notation: Mε (x) denotes ψ(B(0, ε)), the local M -leaf through x. 8.2. The W H -Holonomy Action. Let E ⊂ TX be a subbundle. We say that E is integrable if there exists a Hölder foliation F of X with smooth leaves such that T F = E. Fix a Lyapunov k ±α H hyperplane H ⊂ R , with corresponding foliations W . Let E be the subbundle of TX which is the sum of the tangent bundle to the H-orbits, and the tangent bundles Eβ to the foliations W β, β 6= ±α. k + + β u If a ∈ R is regular, let Ea denote the unstable bundle of a and ∆ (a) = β ∈ ∆ : W ⊂ Wa , + u L β + α L β so that Ea = T Wa = β∈∆+(a) E . If α ∈ ∆ (a), let Ec = β∈∆+(a)\α E be the dynamically defined complementary bundle. Notice that while each Eβ is integrable, Ecα may, and often does, fail to be integrable. We say that W α is the slow foliation for a if α ∈ ∆+(a) and there exists ||(ta)∗|Eα || −1 C > 0, λ < 1 such that ≤ Cλt for all t ≥ 0, where m(A) = A−1 is the conorm of m((ta) | ) ∗ Ecα s α α A (ie, Ea = E ⊕ Ec is a dominated splitting). Notice that a priori, a slow foliation may not exist at all. The following lemma shows that they are guaranteed to exist for elements sufficiently close to a Lyapunov hyperplane.

k + Lemma 8.4. For every α, β ∈ ∆, there exists a regular a ∈ R such that α, β ∈ ∆ (a) and α is the slow foliation of a. If W α is the slow foliation of a, then Ecα is integrable to a foliation Wdα, and if α α α α r y ∈ Wd(x), then the induced holonomy map Hx,y : W (x) → W (y) is C , where r = 2 or ∞ if the dynamics is C2 or C∞, respectively.

Proof. We first prove existence of the element a. Begin by choosing a0 such that a0 belongs to the Lyapunov hyperplane for α, and uniformly expands β. Let a be a small perturbation of a0 which now expands W α. By Lemma 5.4, α is the slow foliation for a. The existence and properties of the foliation Wdα comes from the standard works in partially + hyperbolic dynamics. The definition of slow foliation immediately implies that the splitting Ea = Eα ⊕ Ecα is a dominated splitting for the diffeomorphism a. The existence of the foliation can then be deduced from [59]. The foliation is Cr by the Cr-section theorem of [34] (its application to this setting can be found, for instance, in [40, Proposition 3.9]). The smoothness of the foliation implies that the holonomies are smooth. We know even more about the structure of the holonomy maps, namely that they are linear in the α α system of local forms coordinates ψx for the foliation W defined in Section 3.3. Let mλ : R → R denote multiplication by λ.

α −1 α α Lemma 8.5. Under the same assumptions as Lemma 8.4, (ψy ) ◦ Hx,y ◦ ψx = mλ for some λ ∈ [A−2,A2], where A is as in Lemma 5.4.

k Proof. We establish some basic identities relating the dynamics of R , normal forms and holonomy α k α α maps Hx,y. First, the normal forms coordinates give that for any a ∈ R , z ∈ X, aψz = ψa·z ◦La(z), where La(z): R → R is multiplication by the (norm of the) derivative of a restricted to the α-leaves α 0 at z, since (ψz ) (0) = 1. Notice also that since the stable holonomies are deﬁned through invariant k −1 foliations, for any a ∈ R and x1, x2 ∈ X, aHx1,x2 a = Ha·x1,a·x2 . Furthermore, if xn, yn ∈ X α satisfy yn ∈ Wd(xn), and d(xn, yn) → 0, then Hxn,yn → Id. 45 k + Let a ∈ R be as in Lemma 8.4, and perturb a to a1 ∈ ker α which still contracts every γ ∈ ∆ (a)\ α (this was a in Lemma 8.4). Notice that the sequences L n (x) and L n (y) are multiplications by 0 a1 a1 constants bounded below and above by A−1 and A by Lemma 5.4. By choosing subsequences, we n n nk nk may assume that L k (x) → mλ1 , L k (y) → mλ2 and that a · x, a · y → z for some z ∈ X a1 a1 (notice that the iterates of x and y must converge to the same point since they are in the same leaf α −1 of Wd(x)). Notice that A ≤ λ1, λ2 ≤ A. Putting this all together, we get that:

α −1 α α −1 −nk nk α (ψy ) ◦ Hx,y ◦ ψx = (ψy ) ◦ a H nk nk a ◦ ψx 1 a1 ·x,a1 ·y 1

n −1 α −1 n n α n = La k (y) ◦ (ψank ·y) ◦ H k k ◦ ψank ·x ◦ La k (x) a1 ·x,a1 ·y

→ mλ1/λ2

n n α α since the holonomy maps H k k converge to Id and the normal forms charts ψ nk and ψ nk a1 ·x,a1 ·y a ·x a ·y α both converge to ψz by continuous dependence. The bounds on λ1 and λ2 imply the bounds in the statement of the Lemma. Remark 8.6. Lemma 8.5 is not a consequence of the usual centralizer theorem for the normal forms of dynamical systems since the holonomy maps cannot be extended globally to a ﬁbered extension commuting with the dynamics. It is also crucial to the remaining arguments that the bound on its derivative does not depend on the closeness of x and y along Wdα. k Fix x ∈ X, and let η be a path based at x in the α, −α and R -orbit foliations. Recall that this ±α means that η is a sequence of points x = x0, x1, . . . , xn such that xi+1 ∈ W (xi) or xi+1 = ai · xi k ˜ for some ai ∈ R . If β ∈ ∆ is not equal to ±α, let D(α, β) = D(α, β)∪D(−α, β)\{±α} ⊂ ∆ be the set of weights which can be expressed in the form tβ + sα, for t > 0, s ∈ R. Notice that by picking a ∈ ker α such that β(a) < 0, we may perturb a to a1 so that D˜(α, β) ∪ {−α} is stable for a1 or ±α to a2 so that D˜(α, β) ∪ {α} is stable for a2. Therefore, W can be made the slow foliation for a1 r or a2, respectively, implying that D˜(α, β) integrates to a Hölder foliation with C leaves which we denote F α,β, and that F α,β-holonomy between two α-leaves or two −α-leaves is Cr. β k Lemma 8.7. If β ∈ ∆ satisﬁes β 6= ±α and y ∈ W (x) and η is an α, −α, R -path based at x k 0 as described above, then there exists an α, −α, R -path η based at y with the same number of legs α,β y = y0, . . . , yn such that yi ∈ F (xi). Proof. We lift each leg of η one at a time, proceeding by induction. Assume we have lifted the points x0, . . . , xi to y0, . . . , yi. We now show that we can continue to xi+1 and yi+1. By Lemma γ 5.34, the lifts have the following structure: at xi, we have a distinguished path in the foliations W , γ ∈ D(α, β) such that

W γ1 W γ2 W γm (8.1) xi = xi,0 −−−→ xi,1 −−−→ ... −−−→ xi,m = yi. F where x −→ y means that y ∈ F(x) for the foliation F. First, suppose that xi+1 = a · xi for some k a ∈ R . Then for each leg connecting xi to yi, the a-holonomy is clear:

W γ1 W γ2 W γm xi+1 = a · xi = a · xi,0 −−−→ (a · xi,1) −−−→ ... −−−→ (a · xi,m) = a · yi = yi+1. α Now, suppose xi+1 ∈ W (xi) (the proof for −α is identical). One may make α the slow foliation in a stable leaf containing γi for each γi in the connection (8.1) (as discussed before the statement of the lemma). So by Lemma 8.4, one may use stable holonomy to transport the connections in (8.1). In particular, we get: 46 W [α,γ1 ] W [α,γ2 ] W [α,γm] (8.2) xi+1 =x ˜i,0 −−−−−→ x˜i,1 −−−−−→ ... −−−−−→ x˜i,m =: yi+1.

Here, each x˜i,j is obtained as the image of stable holonomy from x˜i,j−1. Notice that [α, γi] ⊂ D(α, β), by Lemma 5.33, each connection of the form W [α,γm] above can be placed in a circular ordering, and therefore a new connection (possibly with more legs) can be made between xi+1 and yi+1. We then iterate the procedure and get the result by induction. k k Lemma 8.8. Let η be any α, −α, R -path based at x, and a ∈ R . Deﬁne a ∗ η to be the path which starts at x, moves along η, then moves along a. Then if η0 and (a∗η)0 are the paths obtained through the holonomy deﬁned in Lemma 8.7, e((a ∗ η)0) = a · e(η0).

k Proof. Notice that in the proof of Lemma 8.7, any connection x −→R y corresponds to y = a · x. Since this connection is the last type, it is lifted to a connection of the form y0 = a · x0. Since this 0 is the last connection of the path η, the last connection of η also takes this form. The following definition is crucial in the subsequent analysis. In general, the number of legs for a given W H -path may increase after applying a ±α-holonomy. We shall see that another notion, the a-switching number remains constant. Definition 8.9. Fix a ∈ H such that a 6∈ ker β for β 6= ±α, and let ρ be a W H -path in the foliations β W . Then ρ has some combinatorial pattern of weights (β1, . . . , βn). Define the a-switching number of ρ, s(ρ), to be one plus the number of weights βi such that βi(a)βi+1(a) < 0. Corollary 8.10. Let η be a broken path in the W α, W −α and L-orbit foliations based at x, and ρ be H k H a path based at x in W . Then there exists an α, −α, R -path πρ(η) based at e(ρ) and a W -path πη(ρ) based at e(η) such that e(πρ(η)) = e(πη(ρ)) and s(ρ) = s(πη(ρ)). The functions πη and πρ are continuous.

H Proof. Apply Lemma 8.7 inductively along each leg of the W -path. The path πρ(η) is the corre- k sponding α, −α, R path at the end of the process, and the path πη(ρ) is the connections made by the β-foliations. Finally, observe the switching number is preserved by (8.2), since α(a) = 0 and if β(a) < 0, every γ ∈ [α, β] has γ(a) < 0 (similarly for β(a) > 0). H k −2n Lemma 8.11. If ρ is a W -path with s(ρ) ≤ n and η is an α, −α, R -path, then A `(η) ≤ 2n `(πρ(η)) ≤ A `(η), where A is as in Lemma 5.4. 2 Proof. By induction, it suffices to show that πρ(η) ≤ A `(η) when s(ρ) = 1. That is, when there exists a ∈ H such that a contracts every weight in ρ. Recall that πρ(η) is defined through stable holonomy, so the paths πρ(η) and η converge to one another under iterations of a. But since the derivative of a is bounded above by A and below by A−1 when restricted to each α and −α leaf, the lengths of a · πρ(η) and a · η are each distorted by at most A. Thus, since they converge to one 2 another, the lengths of η and πρ(η) differ by at most A . 8.3. Controlling W H -returns to M α. Lemma 8.11 gives us Lipschitz control on W H -holonomies, which we will use to build an equicontinuous action of holonomies. The difficulty in doing this directly is that the derivative bound depends on N, the a-switching number. Therefore, if x ∈ X, H H we define WN (x) to be the set of endpoints of W -paths ρ with s(ρ) ≤ N starting from x.

The Return Dichotomy. For every H-periodic orbit p, and every N ∈ N, we have the following dichotomy: 47 α H α (1) For every x ∈ M (p), WN (x) ∩ M (p) is a ﬁnite set. α H α (2) There exists x ∈ M (p) such that WN (x) ∩ M (p) is inﬁnite. If every N and p fall into case (1) of the dichotomy, we will construct a rank one factor of the action (see Section9). Otherwise, assuming case (2) some N and periodic point p, we will show that W H (x) is dense in X for some x ∈ X (and therefore H has a dense orbit by Corollary 5.25, see Section 10).

H α Lemma 8.12. Let p be H-periodic. There exists N ∈ N such that WN (M (p)) (the saturation of α H M (p) by WN ) is all of X. H α Proof. In fact, we show that W4 (M (p)) = X. Fix a0 ∈ H which is not in any other Lyapunov H α hyperplane, and let a1 be a perturbation of a0 which is regular. First, notice that W2 (M (p)) contains an open neighborhood of M α(p). Indeed, using local product structure, any y ∈ X close α k u to x ∈ M (p) is reached by applying some element b ∈ R , moving along Wa1 (b · x) to some z, s then arriving at y ∈ Wa1 (z). Now, by Lemma 8.4, one may connect b · x to y by ﬁrst moving + along the α-leaf, then along the weights in ∆ (a1) \ α. Similarly, one may isolate the −α-leg in + H the move from y to z. Notice that the weights of ∆ (a1) \ α are all in W . Therefore, one may project the ±α-leaves down to x along these W H -paths, so that we move from x to a nearby point 0 α k s u x ∈ M (p) using α, −α, R -moves, then along Wa0 and ﬁnally along Wa0 . From this, it is clear H α that y ∈ W2 (M (p)). H α k Now, W2 (M (p)) is invariant under the R -action, and contains an open set, so is therefore dense k by transitivity. Repeating the argument above proves that after saturating again by α, −α, R -leaves s/u and Wa0 leaves, we must get all of X, since the size of the neighborhoods is uniform. Since we may k again push the α, −α, R -leaves to the start of the path using holonomy projections, we conclude H α that W4 (M (p)) = X.

9. Finite returns: Constructing a rank one factor In this section, we prove that if Case (1) of the dichotomy above holds for every N and periodic H point p, we may construct a smooth rank one factor of the action. We recall that WN and the number of switches is defined by fixing some a ∈ H such that a 6∈ ker β for every β 6= ±α, so that 0 k 0 a acts partially hyperbolically on X. We may perturb a to a ∈ R which is regular, so that a acts k normally hyperbolically with respect to the orbits of the R -actions. H Lemma 9.1 (Local integrability of W ). Assume Case (1) holds for every N. There exists ε0 with H the following property: suppose that ρ is a W2 -path which begins with its a-stable legs, then moves along a-unstable legs. Then for every ε0 > ε > 0, there exists δ > 0 such that if the legs of ρ all H 0 have lengths less than δ, there exists a W2 -path ρ with the same base and endpoint as ρ, which begins with its unstable legs, then moves along stable legs, and every leg has length at most ε (for both paths, the position of the H-legs do not matter). Proof. We recall some facts from the theory of partially hyperbolic transformations. Recall that a0 ∗ 0 acts normally hyperbolically, and let Wa0,δ denote the local manifolds of a (where ∗ = s, u, cs, cu stands for stable, unstable, center-stable, center-unstable, respectively). Since a0 acts normally u cs hyperbolically, there exists δ > 0 such that the map g : Wa0,δ(x) × Wa0,δ(x) → X defined by cs u g(y, z) = Wa0,δ(y) ∩ Wa0,δ(z) is well defined and a homeomorphism onto a neighborhood of x. Choose ε0 to be such that B(x, ε0) is contained in this neighborhood. Let w ∈ B(x, ε) be reached by first moving along some a-stable legs, then some a-unstable legs. Note that anything which is a-(un)stable is automatically a0-(un)stable. Then we may reverse the 48 u cs order of the stable and unstable legs by writing w = g(y, z) for some y ∈ Wa0,δ(x) and z ∈ Wa0,δ(x). 0 k Notice also that since the center foliation of a is given by an abelian group action of R , we may k reach w from y by first applying an element b of the R action to get to y1 = b · y, then moving s s k along the Wa0 (y1) to get to w (ie, w ∈ Wa0 (y1)). By choosing a subgroup L ⊂ R transverse to H, we may further write b = h + `, where h ∈ H and ` ∈ L. The plan is as follows: We wish to show that the `-component, and the W ±α-components of the s/u Wa0 -paths are trivial. So we turn them into continuous functions depending on the points y and z. If the components are not trivial at some point, they can be made arbitrarily small by continuity and the fact that they are trivial for the trivial path. Each component can be pushed to the end, giving rise to a new intersection point of M α(x) if nontrivial. By the discreteness assumption, we will conclude that the components must be trivial at all points. We now provide the details of the proof. Assume that a0 is chosen sufficiently close to a so that u s α and −α are the slow foliations in Wa0,δ and Wa0,δ, respectively. Thus, by Lemmas 5.33 and 8.4, 0 α 0 −α we may break off the α and −α pieces to get points y ∈ W (x) and w ∈ W (y1) such that u 0 s 0 y ∈ Wa,δ(y ), and w ∈ Wa,δ(w ). Notice that there now only three legs which are not permitted in H 0 0 W : the α leg connecting x and y , the −α leg connecting y1 and w , and the L-leg connecting h · x and y1. One may (in a way which assigns a unique path) use the holonomy projections to arrange all α u three of these legs to appear at x to arrive at a point x1 ∈ M (x), then move along Wa , then along s Wa , then along the H-orbit. Since the holonomies are continuous, and the transverse intersection must vary continuously, the point x1 is a continuous function of y. Call η the α, −α, L-path used to get from x to x1. Let q(w) = d(x, x1). Then q is a continuous function whose domain is the set of points reached from x by an a-stable leaf, then an a-unstable leaf, whose lengths are each at most ε, and q(x) = 0. The lemma, stated in this language, exactly states that q ≡ 0 where defined (ie, on the points reached by a local a-stable leaf, then a local a-unstable leaf). Suppose that q 6≡ 0, ie that there exists w such that q(w) > 0. Choose any path γ : [0, 1] → B(x, ε) such that γ(0) = x and γ(1) = w, and q is defined on γ(t) for every t ∈ [0, 1]. This is possible simply by collapsing the leaves. Then q ◦ γ : [0, 1] → R≥0 is continuous, so by the intermediate value theorem, for sufficiently large n, 1 there exists tn ∈ [0, 1] such that q(γ(tn)) = n . For each n, we use the notations developed above for H α 0 w = γ(tn). Choose any W -path ρ from x to M (p), and let N = s(ρ) and x = e(ρ). Then one may construct paths of length 2N + 4 starting from x0 in the following way: first follow the reverse of ρ to end at x. Then follow the “commutator path” by following the a-stable/unstable path to get to y, then the a-stable/unstable path to get to xn, where xn is the endpoint of the construction above for w = γ(tn) (corresponding to the point x1), and finally end by following πη(ρ), which will N N 0 end at a point whose distance is at most A q(γ(tn)) = A /n from x . This contradicts that we are in Case (1), so we conclude q ≡ 0.

Lemma 9.2. For every N ∈ N and suﬃciently small ε > 0, there exists δ = δ(N) such that if ρ H α is a WN -path based at a point x ∈ X such every leg in ρ has length less than δ and e(ρ) ∈ Mε (x), then e(ρ) ∈ Hx.

u s 3 Proof. We first prove the lemma for N = 2. Define the following map f : Wa (x)×Wa (x)×H ×R → s cu 0 X. Pick (y, z, h, b, t, s) in the domain. Let w = Wa0 (y) ∩ Wa0 (z). Then let w = (h + b)w. α 0 0 −α Let w1 be the point of W (w ) at signed distance t from w and w2 be the point of W (w1) at signed distance s from w1. Setting f(y, z, h, b, t, s) = w2, we get that the map f is a local homeomorphism onto its image (one can rearrange the α and −α legs as in the proof of Lemma 9.1 49 and use the standard local product structure of Anosov actions). In particular, one cannot have f(y, z, h, 0, 0, 0) = f(x, x, 0, b, t, s) unless all moves are trivial. Now we proceed in general. Let ε > 0 be given by the assumptions of Lemma 9.1. We may apply H H Lemma 9.1 to arrive at some δ = ε1 for which any W2 -path which begins with W -stable legs and ends with W H -unstable legs can have its stable/unstable order reversed, and after reversal, the new path has its legs less than ε. Then again apply Lemma 9.1 to ε1 to arrive at a new δ = ε2 < ε1/2 for which rearrangements of paths of length at most ε2 has its lengths at most ε1/2 after rearrangement. Repeat this process N times to arrive at a sequence of εi such that εi < εi−1/2 and after rearranging a stable/unstable path to an unstable/stable path with legs of length at most εi, the new legs have length at most εi−1/2. Then, if every leg of ρ has length at most εN , we may apply Lemma 9.1 N times to rearrange the path to begin with an unstable leg, then move along a stable leg (or vice-versa). Since the lemma holds for N = 2, we have finished the proof. H Lemma 9.3. Fix x ∈ X. For every y ∈ WN (x), there exists δ0 = δ0(y) > 0 such that if z is reached H H from y by a W2 -path whose legs have length at most δ0, then z ∈ WN (x). The constant δ0 depends only on the length of the final group of stable or unstable legs of a W H -path connecting x and y, H and is bounded below in a local W2 -leaf of x. H Proof. Fix any WN -path ρ connecting x and y, and suppose that ρ ends with stable legs, so that ρ = ρ2 ∗ ρ1, where s(ρ1) ≤ N − 1, ρ1 ends with an ustable leaf and ρ2 is contained in a single stable leaf. Then let L denote the length of ρ2. Choose t > 0 such that the length of (ta) · ρ2 is smaller than δ as in Lemma 9.1 applied to some fixed ε. Let δ0 = δ/ ||d(ta)||. Then if ρ3 is any H H W2 -path starting from y whose legs have length at most δ0, (ta) · (ρ3 ∗ ρ2) is a W4 -path whose legs H all have length at most δ. By Lemma 9.1, we may find a W2 path which shares the same start and endpoint, begins with an unstable leaf and ends with a stable leaf. Applying −ta to this path yields the desired path and absorbing the unstable leaf into the last leaf of ρ1 gives the desired path. k r Lemma 9.4. Let R y X be a C action, with r = 2 or r = ∞. For sufficiently large N, the H α r0 0 collection WN (x): x ∈ M (p) is a C foliation of X, where r = (1, θ) or ∞, respectively. Proof. Fix N as in Lemma 8.12, and choose x ∈ X. Then there exists a W H -path ρ connecting x to some x0 ∈ M α(p) with s(ρ) ≤ N. H 0 0 α H 0 By assumption (1) applied to W2N+6(x ), there exists ε > 0 such that if y ∈ M (p) ∩ W2N+6(x ) 0 0 0 0 0 0 α and d(x , y ) < ε, then x = y . Choose δ such that if dX (x, y) < δ and y ∈ Mε (x), then y is k connected to x by an α, −α, R -path whose length is at most ε (this is possible by the continuity of the map from Lemma 8.3). Let 0 < δ < A−N δ0 be chosen so that Lemma 9.2 applies with N = 6. Then define the following map f : B(x, δ) → M α(p)/H. Recall that a ∈ H does not belong to any other Lyapunov hyperplane. If z ∈ B(x, δ), one may use local product structure to find a unique path connecting x and z which moves along W α(x) to a point y, then along W −α(y) to a H point z, then along the L-orbit to a point w, then along a short W2 -path to arrive at z. Project the α, −α, L-path connecting x and w along ρ so that the endpoint after the projection is some z0 ∈ M α(p). Then define f(y) to be Hz0 ∈ M α(p)/H. H Let Fy be the set of points in B(x, δ) reached from y by W2 -paths whose legs have length at −1 most ε. We claim that if y ∈ B(x, δ), then Fy = f (f(y)). Indeed, if y, z ∈ B(x, δ) and there is a H 0 W2 -path ρ connecting y to z, then when one connects x to y and x to z, one obtains intermediate 0 0 α H points y , z ∈ M (x) connected from x by short α, −α, L-paths η1 and η2, respectively, and W2 - 0 0 0 0 paths ρ1 and ρ2 connecting y and z to y and z, respectively. Notice that if y 6= z , then we may 0 −1 H 0 0 move along the path ρ1, then ρ , then ρ2 and get a W6 -path connecting y and z . By choice of δ 0 0 −1 and Lemma 9.2, we conclude that y = z . Therefore, Fy ⊂ f (f(y)). 50 Now suppose that f(y) = f(z). Then the paths η1 and η2 constructed above project to the same α −1 H path on M (p) by Lemma 8.3, so η1 = η2. Therefore, the path ρ2ρ1 is a W4 -path connecting y H −1 and z. Applying Lemma 9.1 to the middle switch implies that z ∈ W2 (y), so f (f(y)) ⊂ Fy. For simplicity of writing, we let r0 = 1, θ or r0 = ∞ depending on whether r = 2 or r = ∞. To H r0 r0 see that WN has the structure of a C foliation, we claim it suffices to show that the map f is a C submersion. Indeed, in this case one can see that since the chosen path connecting points of B(x, δ) α to M (p) varies continuously, the δ0 of Lemma 9.3 can be chosen uniformly in a neighborhood of x. Therefore, in a sufficiently small neighborhood of x, if f is a Cr0 submersion, the preimages f −1(y0) H form a foliation, which we have just shown are the local leaves of WN (y) by Lemma 9.3. We now prove that f is a Cr0 submersion. Fix y ∈ B(x, δ), and consider some z ∈ W α(y). Then f(z) is defined by first moving along a local M α-path from x to some point z0, then along H α H α α 0 0 a WN -path. Notice that since z ∈ W (y), the WN -path connects Mε (y) and Mε (z ), so the y α 0 appearing in the definition for y will belong to MAN ε(z ), so the projection is defined by applying H 0 the holonomies of this WN -path together with ρ which connects x and x . Therefore, we may iterate Lemma 8.4 to obtain that f is Cr0 along W α(y) and W −α(y) for every y ∈ B(x, δ). Fix a k perturbation b ∈ R of a which is not contained in any Lyapunov hyperplane and for which ±α are ±α u/s both slow foliations (recall the start of Section 8.2). This determines foliations W[ ⊂ Wb by Lemma 8.4. Notice that Wdα(x) and W[−α(x) are both Hölder foliations with smooth leaves, and α α u that π collapses each to a point. Since W and Wd are complementary foliations inside Wb (x), f u s r0 is smooth along Wb (x). Similarly, f is smooth along Wb (x). It is also clear that f is C along k s k R -orbits. Applying the Journé theorem to the foliations Wb and R gives that π smooth along cs cs u r0 Wb . Then apply Journé again to the foliations Wb and Wb to see that f is C on B(x, δ). Remark 9.5. The reason for the loss of regularity from C2 to C1,θ occurs in the application of Journé’s theorem. Under the assumption of C2,θ, there is no such loss of regularity. A Cr version of our results will also hold, r ≥ 2.

H H Lemma 9.6. For suﬃciently large N, WN (x) = W (x).

H β Proof. By Lemma 9.4, WN is a smooth foliation containing local W -leaves and H-orbits. Since its α H leaves intersect M (p) in ﬁnitely many H-orbits, it must be codimension 3. Therefore, T (WN ) = L β H TH⊕ β∈∆\{±α} E by the Frobenius theorem. Therefore, any W -path, regardless of its structure, H H H is tangent to WN , so WN (x) = W (x).

H α Lemma 9.7. Each leaf WN (x), x ∈ M (p) is compact.

H H Proof. Suppose that xk ∈ WN (x) converges to some y ∈ X, and let ρk be a WN -path connecting xk and x. If xk enters a fixed, small neighborhood of y (where local product structure applies) H 0 0 we may connect xk and y by a short W2 -path ρk, then a short α, −α, L-path ηk, so that ηk ∗ ρk 0 0 is a path connecting xk and y. Hence ηk ∗ ρk ∗ ρk and η` ∗ ρ` ∗ ρ` are paths connecting x and y. −1 0 −1 −1 0 So ρ` ∗ ρ` ∗ η` ∗ ηk ∗ ρk ∗ ρk is a path beginning and ending at x. Applying the holonomies 0 −1 H k of ρk and ρk to ηk,` = η` ∗ ηk yields a W2N+4-path ρk,` and a short α, −α, R -path ηk,`, both N+2 based at x, with the same endpoints. Since the length of ηk,` is at most A times the length −1 of η` ∗ ηk, and the lengths of ηk and η` both tend to 0, from the assumption of Case (1) applied H to W2N+4, we conclude that ηk = η` for sufficiently large `. Since the sequence ηk also converges to the trivial path, we conclude that ηk is eventually the trivial path. Therefore, by Lemma 9.6, H H y ∈ WN+2(x) ⊂ WN (x). 51 α Recall that Mε (x) is the image of B(0, ε) under the charts ψ defined in Lemma 8.3 at the point x. Given x ∈ X, let r(x) denote the cardinality of (W H (x) ∩ M α(p))/H.

Lemma 9.8. There exists r0 such that r(x) ≤ r0 for all x ∈ X, and there exists an open, dense set α α U ⊂ M (p) (in the topology of M (p)) such that r(x) = r0 on U. Proof. First, notice that r is always finite, by Lemma 9.6 and the assumption that we are in Case (1) for every N. We claim that r is lower semicontinuous. Indeed, suppose that r(x) = c. Then H α one may choose WN -paths ρ1, . . . , ρc starting from x and ending at M (p) and ending at distinct α α α points y1, . . . , yc ∈ M (p). Let ε be such that Mε (yi) ∩ Mε (yj) = ∅ for all i 6= j, and consider the α H saturation of MA−N ε(x) by local W -leaves, call this set V . Then projecting the paths ρi to these 0 0 0 0 nearby points x ∈ V yields c paths ρi with distinct endpoints yi. Therefore, r(x ) ≥ r(x) whenever x0 ∈ V , and r is lower semicontinuous. Now, every lower semicontinuous function on a compact space takes a maximum value, call it r0. k Since it takes discrete values, r(x) = r0 on an open set. Furthermore, r is invariant under the R H action, so r(x) = r0 on an open dense set in X. Since it is also saturated by W -leaves, r = r0 on α an open and dense subset of M (p), as well. H α k Lemma 9.9. Let ρ be a W -cycle based at a point x ∈ M (p), and η be an α, −α, R -path. Then πη(ρ) is also a cycle. Proof. Notice that it suffices to prove the lemma when η consists of a single leg, since by definition, k πη is the composition of the projections of the legs of η. If η is only a piece of an R -orbit, the k H H lemma follows from the fact that the R -action takes W -cycles to W -cycles. We prove it here when η is a single α leg (the proof for −α legs is identical). Notice that πη(ρ) is a cycle if and only if α α e(πρ(η)) = e(η). That is, it suffices to show that πρ : W (x) → W (x) is the identity map. Notice n that from Lemma 8.5, πρ is an affine map which fixes x. If πρ 6= Id, then the orbits πρ (y): n ∈ Z n H H H are infinite. Since every πρ (y) ∈ W (y), and W (y) = WN (y) by Lemma 9.6, we arrive at a contradiction to Case (1) of the return dichotomy. This finishes the proof. Lemma 9.10. The function r is constant on M α(p). α Proof. Suppose that r(x) < r0 for some x ∈ M (p). Then since r = r0 on an open dense subset of α 0 0 k M (p), there is a nearby point x such that r(x ) = r0. Choose a short α, −α, R -path η connecting x and x0. Then project the set of paths giving the intersection point of W H (x0) ∩ M α(p) to x. Since the cardinality is strictly greater, there must be some W H -path ρ0 starting from x0 and returning 0 0 0 H to some y 6= x such that ρ = πη(ρ ) is a cycle at x. Then πη−1 maps a W -cycle ρ to a non-cycle, α contradicting Lemma 9.9. So r is constant on M (p). k r H r0 Proposition 9.11. If R y X is C , r = 2 or r = ∞, then the space X/W is a C manifold, H r0 k and the projection π : X → X/W determines a C , non-Kronecker rank one factor of R y X, where r0 = (1, θ) or ∞, respectively. Proof. We model the space X/W H locally on M α(p)/H. Notice that by Lemma 8.12, every W H - leaf intersects M α(p), and by Lemma 9.10 the number of intersections is constant. We construct α H α manifold charts as follows: given x := x1 ∈ M (p), let {x1, . . . , xr} = W (x) ∩ M (p). Choose α α −N α ε > 0 such that Mε (xi) ∩ Mε (xj) = ∅ if i 6= j. Let δ = A ε and let U = Bδ(x) ∩ M (p)/H. Then 3 U is an open set in a 3-manifold, and has a chart ψ : V → U, where V ⊂ R . Define the chart for X/W H by ψ˜(t) = W H (ψ(t)). ˜ H H We first prove that ψ is injective. Any W -path connecting x to xi can be achieved in WN by α Lemma 9.6. Notice that by choice of δ, each point y := y1 ∈ U must intersect M (p) in points {y1, . . . , yr} near {x1, . . . , xr}, so y1 is the only yi in U. Therefore, the map is injective. 52 The transition maps between charts are Cr0 , since W H is a Cr0 foliation. Therefore, X/W H has an induced Cr0 manifold structure, and the projection X → X/W H is Cr0 by construction. Finally, k H H H since the R -action takes W -leaves to W -leaves, and H fixes the W -leaves, the action descends H to an Anosov flow on X/W .

10. Infinite returns: Finding a dense H-orbit

Throughout this section, we assume that we are in Case (2) for some N ∈ N and periodic point p. While in Case (1), we showed the existence of a rank one factor, in this section we show that we have one of the other two conclusions of Theorem 2.1: that there is a dense H-orbit or that H-orbit closures are dense in the ﬁber of some circle factor (Proposition 10.14). Recall the holonomy action of W H -paths on M α paths (and vice versa) given by Corollary 8.10: H k if ρ is a W -path and η is an α, −α, R -path which share the same base point, πρ(η) is the path η “slid along” ρ, beginning at e(ρ), and similarly πη(ρ) is the path ρ “slid along” η, beginning at e(η). We ﬁrst prove the following:

α H α H α Lemma 10.1. For any x ∈ M (p) such that WN (x)∩M (p) is infinite, x ∈ W2N (x) ∩ M (p) \{x}. H Proof. By assumption, there are infinitely many WN -paths ρ1, ρ2,... based at x such that xi = e(ρi) are all distinct. Since {xi} are infinitely many distinct points in a compact space, they accumulate somewhere, and for every ε > 0, there exist i and j such that 0 < d(xi, xj) < ε. k Choose a very short α, −α, R path ηij connecting xi and xj, and let ìj = ìj(ε) denote its −1 H length. Notice that ìj → 0 as ε → 0. Recall ρi is the opposite W -path connecting xi with x. 0 −1 H Then ρi = πηij (ρi ) is a W -path connecting xj with some point whose distance from x is at most N 0 0 −1 A ìj by Lemma 8.11. Furthermore, s(ρi) = s(ρi) ≤ N, so the concatenation ρi ∗ ρj connects x N 0 −1 to a point at distance at most A ìj from x, and has s(ρi ∗ ρj ) ≤ 2N. Fix L as in Lemma 8.3. We say that a W H -path ρ is a local M α-return at x if it is a W H -path k based at x, and there exists an α, −α, R -path η of length at most L such that e(ρ) = e(η). Then α set SN to be the set of points x ∈ X such that there exists a sequence of local M -returns ρn at x such that e(ρn) → x and e(ρn) 6= x. α k Lemma 10.1 exactly implies that there exists x ∈ SN ∩ M (p) for some R -periodic orbit p and H α some N ∈ N. Notice that for any point y ∈ W (x)∩M (p) sufficiently close to x, y is in the image of k+2 −1 H α ψx, where ψx is as in Lemma 8.3. Let Ax ⊂ R be defined by Ax = ψx (W (x)∩M (p)∩B(x, ε)). H α Notice that since we assume that x ∈ W (x) ∩ M (p), 0 ∈ Ax. That is there exists a sequence 2 2 2 (tn, sn, vn) → 0 such that tn + sn + ||vn|| → 0. We describe types of accumulation for elements of SN , which we call Type I-V (in each case, we allow the ability to take a subsequence):

I. Generic accumulation tk, sk 6= 0 and vk 6∈ H for all k II. Center-stable accumulation For every k, sk = 0, vk 6∈ H and tk 6= 0 (or tk = 0, vk 6∈ H and sk 6= 0) III. Orbit accumulation For every k, sk = 0 and tk = 0 IV. Orbit-less accumulation For every k, vk ∈ H and sk, tk 6= 0 V. Strong-stable accumulation For every k, sk = 0 and vk ∈ H (or tk = 0 and vk ∈ H) We will prove the following Lemma for each of the cases above:

α Lemma 10.2. For every N ∈ N, there exists ε0 > 0 and M ∈ N such that if x ∈ SN ∩ M (p) and α −α k has an accumulation of type I,II,III or IV, and y ∈ Wε0 (x) ∪ Wε0 (x) ∪ R x, then y ∈ SM . 53 k In each of the cases, since SN is R -invariant, so we only need to show the statement for y ∈ ±α Wε0 (x). α Proof of Lemma 10.2, Type I. By symmetry, it suffices to prove it for y ∈ Wε0 (x). Suppose xk = ψ(tk, sk, vk) → x is a sequence converging to x of type I. Notice that since tk.sk, vk 6= 0 for all k, we may assume every coordinate is strictly decreasing by taking a subsequence. We claim that we may α assume that xk 6∈ Wε0 (x`) for all k 6= `. Indeed, since all three coordinates are nonzero for all k k α but must converge to 0, xk cannot lie on R ⊕ W (x). Since any other local α-leaf stays a positive α α distance from x, Wε0 (xk) must change infinitely often. Let η denote the path in W (x) from x to H 0 y with only one leg. Let ρk be the W -path which connects x to xk, and set ρk = πη(ρk). Define yk = e(πη(ρk)). α α α Notice that by Lemma 8.11, yk ∈ W N (xk) since y ∈ W (x). Hence, since xk 6∈ W (x`), A ε0 ε0 ε0 α yk 6∈ W N (y`). In particular, yk 6= y`. Notice also that if y is sufficiently close to x, then yk stays A ε0 within a bounded distance of y, and therefore has a convergent subsequence, yk → z. As in the proof of Lemma 10.1, we now choose yk and y` such that d(yk, y`) < δ. Then connect them by k 0 0 −1 0 a short α, −α, -path η which begins at y and ends at y . Set ρ˜ = π 0 (ρ ) ∗ ρ . Then R k,` k ` k,` ηk,` k ` 0 −1 0 ρ˜k,` is a path which starts at y, and ends at e(πη0 ((ρ ) )) = e(π 0 −1 (η )), which has length less k,` k ρk ` than AN δ. Furthermore, the endpoint cannot be equal to y since it lies on a distinct α leaf by construction. Thus, since δ was arbitrarily small, y ∈ S2N . α Proof of Lemma 10.2, Type II. Let y ∈ Wε (x), and xk = ψ(tk, 0, vk) → x, vk 6∈ H. Choose a line 0 ∼ L transverse to H, and write vk =v ¯k + H, with v¯k ∈ L = R. Without loss of generality, we may α assume v¯k is strictly decreasing, so that xk 6∈ W (x`) for all k 6= `. We now repeat the argument as in Type I (the assumption that we approach along distinct local α leaves is sufficient for the k argument). The case of −α follows from an identical argument, since the R -components can be assumed to be distinct modulo H. Proof of Lemma 10.2, Type III. The argument is again as in Type I and Type II: in this case, we k must converge along distinct points of the R -orbit, which have distinct α and −α leaves, so we may follow the proof of Type II. Proof of Lemma 10.2, Type IV. The proof is identical to the proof of Type I, since the α and −α- leaves still change infinitely often in the accumulation.

Let Aff(R) = {ax + b : a, b ∈ R, a 6= 0} be the group of affine transformations of R. We think a b of Aff( ) as the matrix group : a, b ∈ , a 6= 0 , whose Lie algebra is Lie(Aff( )) = R 0 1 R R a b : a, b ∈ . The following lemma is a standard result. 0 0 R

Lemma 10.3. For every X ∈ Lie(Aff(R)), there exists g ∈ Aff(R) such that Ad(g)X is a multiple 1 0 0 1 of either or . Any one-parameter subgroup of Aff( ) is either the additive group 0 0 0 0 R {x + b : b ∈ R}, or conjugate to the multiplicative group {ax : a ∈ R+}. α Lemma 10.4. Let x ∈ SN ∩ M (p) and have a type V accumulation (so that there exists xk ∈ H α α WN (x) ∩ W (x) such that xk → x, xk 6= x). There exists ε0 > 0, M ∈ N and z ∈ W (x) \{x} −α α k α such that if y ∈ Wε0 (x) ∪ (Wε0 (x) \{z}) ∪ R x, y ∈ SM . Furthermore, each y ∈ Wε0 (x) \{z} also has an accumulation of type V along the α-leaf. 54 H Proof. Since W is saturated by H-orbits, we may without loss of generality assume that vk = 0. Notice that accumulations of Type V correspond to accumulating along an α or −α leaf. We without loss of generality assume that xk accumulates to x along an α-leaf. In this case, notice that each xk belongs to a distinct local −α leaf, so we may apply the argument from Type I to get y ∈ S2N −α for all y ∈ Wε0 (x). α So we must show y ∈ S2N for all y ∈ W (x). The previous argument fails exactly because the H α leaf fails to change. Notice that since x ∈ SN , there exists a sequence of paths ρn in W with α s(ρn) ≤ N such that e(ρn) ∈ W (x) and e(ρn) → x. We may extend each ρn to a holonomy as α described in Lemmas 8.4 and 8.5. Since e(ρn) ∈ W (x), we may recenter the normal forms charts to get that the endpoints of the path translated along α are given by affine maps ϕn ∈ Aff(R) such −2N 2N that ϕn(t) = ant + bn, A ≤ an ≤ A , bn 6= 0 and bn → 0. −1 Notice that ϕn is a precompact family of linear maps. Therefore, the collection ϕn ◦ ϕm accu- −1 mulates at Id ∈ Aff(R). Furthermore, ϕn ◦ ϕm is a map which associates to a point y, the endpoint of a path in W H based at y with at most 2N switches. Since these maps accumulate nontrivially at the identity, evaluation of these maps at y ∈ W α(x) gives a sequence of points converging to y H which lie in W2N (y). If this accumulation occurs along a sequence without a common fixed point (ie, they do not lie in a multiplicative one-parameter subgroup as described in Lemma 10.3), one α can use the corresponding returns to get that y ∈ S2N for all y ∈ W (x). Otherwise, there is a common fixed point of the transformations converging to Id (this is the point z as in the statement), α and all y ∈ W (x) \{z}, y ∈ S2N and has accumulation of type V along the α-leaf. α 0 Lemma 10.5. If SN ∩ M (p) 6= ∅, then SN 0 contains an open set in X for some N ∈ N. α Proof. We prove the Lemma in two distinct cases: since SN ∩ M (p) is nonempty, we have either α α that every x ∈ SN ∩ M (p) has an accumulation of type I-IV, or there exists x ∈ SN ∩ M (p) which has a type V accumulation. If every point has an accumulation of type I-IV, then one can saturate α −α k α x with local W -leaves, W -leaves and R -orbits to get an set in SN 0−2 ∩ M (p) which is open in α 0 M (p) for some N ∈ N. Notice that we can also saturate with β-leaves by applying β-holonomies to get that for some open set U ⊂ X every point in U is self-accumulated by a W H -path with at 0 k most N switches whose endpoints are connected by arbitrarily short, nontrivial, α, −α, R -paths. This proves the Lemma in this case. α Now assume that x ∈ SN ∩ M (p) has an accumulation of type V, and without loss of generality, α α assume it is along the W -leaf. Then by Lemma 10.4, there is a neighborhood U1 ⊂ W (x) containing x such that every y ∈ U1 also has a type V accumulation. Therefore, again by Lemma k 10.4, we may saturate U1 with −α leaves and still lie in SN 0−2. Since SN 0−2 is saturated by R - α orbits, we have constructed an open set in M (p) ∩ SM . Repeating the argument above obtains an open set of points in X which has the desired self-accumulation property. As an immediate corollary of Lemma 10.5 and Theorem 5.8, we get the following:

Lemma 10.6. There exists N > 0 and ε1 > 0 such that if p is an ε1-dense periodic orbit, p ∈ SN . Let p be a periodic orbit guaranteed by Lemma 10.6. Notice that for any point y ∈ W H (p)∩M α(p) k+2 suﬃciently close to p, y is in the image of ψ, where ψ is as in Lemma 8.3. Let Ap ⊂ R be deﬁned −1 H α H α by Ap = ψ (W (p) ∩ M (p) ∩ B(p, ε)). Notice that since we assume that p ∈ W (p) ∩ M (p), 0 ∈ Ap. k Lemma 10.7. For every R -periodic orbit p belonging to SN , at least one of the following holds: k (a) W H (p) ⊃ R · p (b) W H (p) ⊃ W α(p) 55 (c) W H (p) ⊃ W −α(p)

Since p ∈ SN , we know that p has accumulation of some type I-V described above. We prove that at least one of (a), (b) or (c) holds in each of the cases. The numbering of types I-V was made so that the proof of Lemma 10.2 proceeded in a linear fashion. While the accumulation types of Lemma 10.7 are the same, the proof proceeds more naturally in a different order: III, V, II, I, IV. Proof of Lemma 10.7, Type III. In type III, we know that W H (p) accumulates to p only along the k k H R -orbit of p. We claim that Γ = a ∈ R : a · p ∈ W (p) is a subgroup. Indeed, if a ∈ Γ, there exists a W H -path ρ starting from p such that e(ρ) = a·p. Then (−a)·ρ is a W H -path starting from a−1 · p and ending at p. Reversing the path results in a path from p to (−a) · p. That is, (−a) ∈ Γ. Now suppose that a, b ∈ Γ. Then there exist paths ρ1 and ρ2 beginning at p and ending at H a · p and b · p, respectively. Then a · ρ2 is a W -path that begins at a · p and ends at (a + b) · p. Concatenating ρ1 and a · ρ2 gives a path from p to (a + b) · p, so a + b ∈ Γ. k Now, Γ is a subgroup of R containing the codimension one subgroup H and which accumulates on 0 transversally to H. Therefore, it must be dense. So we are in case (a) of Lemma 10.7. H Proof of Lemma 10.7, Type V. In this case, WN (p) accumulates to p only along an α or −α leaf. Assume without loss of generality that it is along the α leaf, and set C = W H (p) ∩ W α(p). Fix the normal forms parameterization of W α(p) as described in Section 3.3. Let ρ be a W H -path with endpoint e(ρ) ∈ W α(p). Recall from the proof of Lemma 10.2, Type V, that ρ induces a α α map ϕρ : W (p) → W (p) such that the endpoint of πρ([p, x]) is ϕρ(x), where [p, x] is the path α with one leg along W connecting p and x. Furthermore, the map ϕρ is affine in the normal forms coordinates. We have that there exist paths ρn such that ϕn(x) = ϕρn (x) = anx + bn is a sequence −2N 2N of affine maps such that A ≤ an ≤ A and bn → 0. Let Γ be the subgroup generated by ϕn. Notice that Γ is a Lie group and it cannot be discrete since it accumulates at 0. If Γ is 2-dimensional, it must be the identity component of Aff(R), and therefore acts transitively. In particular, we know we are in case (b) of the Lemma. If Γ is 1-dimensional, its identity component is a one-parameter subgroup, and by Lemma 10.3 is therefore either the additive group {x + b : b ∈ R} or conjugate to the multiplicative group {ax : a ∈ R+}. If it is the additive group, 0 has a dense orbit and we are again in case (b) of the Lemma. If it is the muliplicative group, p cannot be the common fixed point since ϕn(p) 6= p for every n. Therefore, the orbit closure of p under Γ is a ray (t, ∞), where t < 0 if the coordinates k are centered at p. Choose an element a ∈ R such that a · p = p and α(a) > 0. Then the orbit of Γ is also invariant under multiplication by a. Therefore, the orbit cannot be a ray, so we know that Γ contains the additive group. So we are in case (b) of the lemma. Proof of Lemma 10.7, Type II. The proof is similar to Type III, but with an additional twist. Let k α H Γ = a ∈ R : W (a · p) ∩ W (p) 6= ∅ . Notice that Γ accumulates at 0 by construction, so it is not discrete. We claim that Γ is a subgroup. Indeed, if ρ is a W H -path that connects p to some x ∈ W α(ap), then a−1ρ is a path that connects a−1 ·p to some y ∈ W α(p). Applying an α holonomy to this path connects some point z ∈ W α(a−1p) to p. Reversing the path gives that a−1 ∈ Γ. H Now, assume that a, b ∈ Γ. Then there exist W -paths ρ1 and ρ2 starting at p and terminating α α at some points x1 ∈ W (ap) and x2 ∈ W (bp), respectively. Then a · ρ2 is a path that begins at ap and ends at some y ∈ W α((a + b)p), and again applying an appropriate α-holonomy, we find a 0 α 0 path ρ1 beginning at x2 and ending at some z ∈ W ((a + b)p). Concatenating ρ2 and ρ1 gives a desired path. We claim that for every a ∈ Γ, ap ∈ W H (p). Indeed, notice that if a ∈ Γ, there exists a path ρ α n such that e(ρ) ∈ W (ap). Choose some a0 such that α(a0) < 0 and a0p = p. Then a0 ρ is still a 56 α n path that begins at p and ends at some point in W (ap). But limn→∞ e(a0 ρ) → ap. Therefore, ap ∈ W H (p), and since Γ is a dense subgroup, we are in case (a) of Lemma 10.7. The last two cases are the most difficult, and we adapt the argument used in Type II to the case when W H (p) accumulates on p along W α(p), but not W H (p)) itself. We develop a lemma which H aids us in the proof. Recall that if ρ is any W -path based at p, ρk determines a holonomy map α α H πρ : W (p) → W (e(ρ)) (see Lemmas 8.4 and 8.5). Say that a sequence of W -paths (ρk) based at p α is a controlled return if s(ρk) is uniformly bounded in k and e(ρk) converges to some point x ∈ W (p) α α and the derivatives of the holonomy maps πρk on W converge. Then let Γ ⊂ Homeo(W (p)) be the set of transformations which are limits of the maps πρk . Lemma 10.8. If f ∈ Γ, then f is affine in the normal forms coordinates at p. Furthermore, Γ is a subgroup of Aff(R). Proof. That each f is affine follows from the continuity of the normal forms coordinates and convergence of derivatives (see Section 3.3). So we wish to show that Γ is a group. Suppose that (ρk) and (σk) are two controlled returns, and that f and g are their corresponding affine maps. Using the local product structure of the coarse Lyapunov foliations and α-holonomies, we may assume α α without loss of generality that e(ρk), e(σk) ∈ M (p). Since e(ρk) → x = f(p) ∈ W (p), we may k 0 construct an α, −α, R -path ηk = ηk ∗ [p, x] as follows: begin with an α-leg connecting p to x (this k 0 is the leg [p, x]), then connect x to e(ρk) by a very short α, −α, R -path (the path ηk). Since e(ρk) converges to x in M α(p), we may do this using the local chart ψ at x as described in Lemma 8.3.

Let σ˜k = πηk (σk) be the projection of the path σk along ηk as built in Corollary 8.10. Then σ˜k 0 0 is a path which begins at e(ηk) = e(ρk). Notice that if σk = π[p,x](σk), then e(σk) = e(π[p,x](σk)) = 0 e(πσk ([p, x])) → g(x) = g(f(p)) as ` → ∞ by definition. Furthermore, since ηk can be made 0 arbitrarily short and s(σk) ≤ N the length of πσk (ηk) can also be made arbitrarily short by Lemma 8.11. Therefore, e(˜σk ∗ ρk) → g(f(p)) so σ˜k ∗ ρk is a controlled return. One can easily check that the derivative of the corresponding limiting holonomy map is f 0(p) · g0(p). Combining these facts exactly implies that the limiting holonomy map is f ◦ g, so Γ is closed under composition. −1 We now check that if f ∈ Γ, then f ∈ Γ. Indeed, suppose that ρk is a controlled return whose 0 limiting holonomy map is f with f(p) = x. Build a path ηk = ηk ∗ [p, x] as before, which begins at k p, moves along the α-leaf to x, then moves from x to e(ρk) with a very short α, −α, R -path. Let ρ¯k and η¯k denote the reversal of the paths ρk and ηk, respectively, so they are both paths connecting e(ρk) to p. Let σk = πη¯k (¯ρk), so that σk is a path based at p. Again notice that since ρk has s(ρk) 0 uniformly bounded, the length of πρk (ηk) can be made arbitrarily small by choosing k very large. −1 In particular, σk is a controlled return. We claim that the limiting map of σk is f . Indeed, we have shown above that σ˜k ∗ ρk is the controlled return which determines the composition. But by construction, σ˜k is the reversal of ρk. Therefore, their corresponding holonomy maps are inverses, and the sequence of maps induced by σ˜k ∗ ρk is constantly equal to the identity. Proof Lemma 10.7, Types I and IV. In this case we will show that we are in case (b) (in fact, we will be able to show both (b) and (c)). We use the local chart near p determined by ψ used to describe cases I-V. We assume that our accumulating sequence xn = (tn, sn, vn) of endpoints of H paths in WN has vn chosen in some fixed one-parameter subgroup L transverse to H, so that we 3 may think of (tn, sn, vn) as a sequence in R . Without loss of generality, because we work locally, α we may assume that tn and sn are the normal forms parameterizations of the leaves W (p) and W −α(p). Define the following cones:

CK = {(t, s, v): |t| > K(|s| + |v|)} 57 K K K K K We claim that for every K > 0, there exists xn = (tn , sn , vn ) ∈ CK such that xn → 0. Indeed, fix some a0 such that α(a0) > 0 and a0p = p. Then apply a0 k times to a path ρ starting at p and H k (k) ending at xn = (tn, sn, vn) in WN so that the new path ends at (λ tn, µ sn, vn), where λ > 1 is (k) k α the derivative of a0 at p restricted to the α, and µ < 1 is the derivative of a0 at ψtn (p) (notice k (k) K k (k) that λ is a power since p is fixed, but µ is only a cocycle). Let xn be (λ tn, µ sn, vn), where K k is the smallest integer such that xn ∈ CK . K k (k) We claim that xn still converges to 0 as n → ∞. Indeed, notice that for (λ tn, µ sn, vn) ∈ CK , k (k) k we must have that λ tn > K(µ sn + vn). The smallest such k will therefore result in λ tn ≤ (k) (k) k λK(µ sn + vn). Since µ < 1, and since sn, vn → 0, we conclude that λ tn → 0 as well. H Define a new set B, which is the set of all points x such that there exists N ∈ N and W - α paths ρk such that s(ρk) ≤ N with e(ρk) ∈ M (p), e(ρk) → x. We claim that p ∈ B \{p}. Indeed, using the normal forms coordinates for W α(p) centered at p, decompose the segment (0, 1] S∞ −(k+1) −k K as (0, 1] = k=0(λ , λ ]. By the existence of the points xn , there exists infinitely many k −(k+1) −k with some xk = (tk, sk, vk) ∈ CK and tk ∈ (λ , λ ]. We claim it is true for all k. Fix k0, and k−k0 −(k+1) −k notice that if k > k0, a0 xk is a point of CK with tk ∈ (λ , λ ]. Letting K → ∞ shows that there is some point of B ∩ (λ−(k0+1), λ−k0 ]. Therefore, B accumulates at p. Notice that B is contained in the orbit of the group Γ defined in Lemma 10.8, so Γ is not discrete. Proceeding exactly as in Type V shows that Γ¯ must be either all of Aff(R) or the additive group. α In either case, W (p) is contained in its orbit, and we are in case (b). 1 Choose a sequence of periodic orbits pn such that pn is n -dense. Then at least one of the cases (a), (b) or (c) of Lemma 10.7 occur infinitely often. Clearly, if a case occurs infinitely often, it must occur on a dense set of periodic orbits. Since (b) and (c) are symmetric we treat only case (b). Lemma 10.9. If (a) occurs on a dense set of periodic orbits, there is a point x ∈ X such that W H (x) = X.

k Proof. Let Y ⊂ X denote the set of points y such that W H (y) ⊃ R · y. Our assumption implies that Y is dense. We claim that it is also a Gδ set, in which case it must contain a point such that k R · y is dense. This will imply the result. k So we must show that it is a Gδ set. Fix a ∈ R , and let Um(a) be the set of all points x ∈ X H 1 such that there exists a W -path ρ based at x such that d(e(ρ), a · x) < m . We claim that Um(a) is open. Indeed, since we may follow a path ρ0 based at y with the same lengths of the legs of the paths of ρ for some fixed Riemannian metric. Since the foliations are continuous, e(ρ0) can be made arbitrarily close to e(ρ). Similarly, by choosing y sufficiently close to x, a · y can be made arbitrarily close to a · x. Therefore, the set Um(a) is open. Finally, observe that the set of points such that a · x ∈ W H (x) is exactly T U (a). Since k m∈N m R k has a countable dense subset, the set of points such that the R -orbit is contained in W H (x) is also a Gδ set. k Given p, q ∈ X periodic, let Λp = StabRk (p) denote the periods of p (the elements a ∈ R which k k fix p). Given a periodic point p, let Bp ⊂ R denote a fundamental domain for R /Λp. k k Lemma 10.10. If (b) holds at an R -periodic point p and q is another R -periodic point, then there H H exists b ∈ Bp such that b · q ∈ W (p). In particular, Bp · W (p) = X. k Proof. Fix a regular element a ∈ R such that −α(a) < 0 < α(a). We may choose a such that there exists x ∈ X with {(ta) · x : t ∈ R+} dense in X (this is possible by Corollary 5.23). Choose some u cs intersection z ∈ Wa (p) ∩ Wa (x). We may connect p to z by first moving along an α leg to arrive at 58 a point y, then along the remaining β-legs expanded by a. We then connect z to x by first moving 0 k along the β-legs contracted by a (other than −α) to a point y , then a −α leg, then the R -orbit. k k We may assume that the R -orbit piece is trivial, since any other point on the R -orbit of x also has a dense forward orbit under a. Call ρ the W H -path connecting y and y0. Now apply ta to this picture, t > 0. By taking a subsequence, we may assume (tna) · x → q and (tna) · p → b · p for some b ∈ Bp. Also, since (b) holds for p, it also holds for (tna) · p, so H 0 (tna) · y ∈ W (tna · p). Note that (tna) · y and (tna) · y are connected by the path (tna) · ρ, and H that s((tna) · ρ) = s(ρ). We may approximate (tna) · y arbitrarily well by endpoints of W -paths based at (tna) · p by the assumption (b). Choose such a path, whose closeness is to be determined k later, and let zn denote its endpoint. Choose a very short α, −α, R -path ηn connecting zn and (tna) · y. Then we may push the path (tna) · ρ along ηn using holonomies, and since the number of switches of (tna) · ρ remains constant, the distance between the new and old endpoints is Lipschitz k controlled by the length of ηn. Finally apply some sn ∈ R so that (sn + tna) · p = b · p to get a path based at b · p (since (tna) · p → b · p, we may assumesn → 0). Notice that by picking all such 00 0 H choices sufficiently small, we get that y = limn→∞(tnv) · y = q belongs to W (b · p). Since the H k −1 W -sets are R -equivariant, b q ∈ W H (p). We now assume that case (b) holds for the remainder of the section. Fix a periodic point p for which Lemma 10.10 holds, and let B = Bp be the corresponding compact fundamental domain for k the R -orbit through p. Call a point x good if B · W H (x) = X, and call G ⊂ X the set of good points. Notice that the set of good points is nonempty since p ∈ G, and therefore dense, since if y ∈ B · W H (p), then y is also good. S S β H If U ⊂ X is any set, let U1 = x∈U H · x ∪ β6=±α W (x) be the first W -saturation of U. H H We inductively define the nth W -saturation of U, Un, to be the first W saturation of Un−1. The H S∞ full W -saturation of U is defined to be U∞ = N=1 UN .

Lemma 10.11. G is a Gδ set. H S∞ H Proof. Recall that W (x) = N=1 WN (x) (it is exactly the full saturation of {x}). Fix a countable dense subset {x } ⊂ X. Then the set of points such that B · W H (x) = X is exactly the set of n n∈N H points such that for every m, n ∈ N there exists a path ρ with legs in W based at x and b ∈ B such that d(b · e(ρ), xn) < 1/m. Notice that this is equivalent to saying that there exists b ∈ B such −1 H H that b · x is in the full W -saturation of B(xn, 1/m). Notice also that the ﬁrst W -saturation of H H any open set is open, so the full W -saturation is as well. Let Um,n be the full W -saturation of B(xn, 1/m), so Um,n is open. Then the set of good points is:

∞ \ −1 B · Um,n m,n=1 In particular, it is a Gδ set. We know from Lemma 10.10 that G is nonempty and dense, and therefore residual. By Lemma 5.24, the set R of points x such that H · x = W H (x) is also residual, as is the set D of points with k k a dense R orbit. Therefore, we may choose x0 ∈ G ∩ R ∩ D. If x ∈ X, let Ax ⊂ R be deﬁned by:

n k o Ax = a ∈ R : a · x ∈ H · x

Lemma 10.12. Ax0 is a closed subgroup containing H as a proper subgroup. 59 −1 Proof. We ﬁrst show that if a ∈ Ax0 , then a ∈ Ax0 . Since a ∈ Ax0 , there exists hn ∈ H such that H hn ·x0 → a·x0. Using local coordinates we can connect hn ·x0 to a·x0 using a W -path with at most k 2 switches, then an α, −α, R -path such that the length of every leg tends to 0. Using β-holonomies and recalling Lemma 5.4, we can rearrange this connection as a path from x0 to a · x0 as follows: k the path begins with an arbitrarily short α, −α, R -path, then moves along a possibly very long H −1 hn ∈ H, then along a short W -path with two switches. Apply a to this path. The derivative of −1 k a is ﬁxed, so we get a path from a ·x0 to x0 which begins with an arbitrarily short α, −α, R -path, H then moves along a possibly very long hn ∈ H, then along some W -path. Reversing the start and −1 H end point of this path shows that a · x0 ∈ W (x0) = H · x0, as claimed.

Now, let a, b ∈ Ax0 . To see that a + b ∈ Ax0 , let hn · x0 → a · x0 and kn · x0 → b · x0. Since b is uniformly continuous, (b+hn)·x0 → (a+b)·x0. Fixing n, we get that (km +hn)·x0 → (b+hn)·x0. Since (b + hn) · x0 can be made arbitrarily close to (a + b)x0, one may choose m and n so that km + hn is arbitrarily close to (a + b) · x0. Therefore, (a + b) · x0 ∈ Ax0 . Finally, notice that it obviously contains H since H · x0 ⊂ H · x0. Furthermore, by choice of x0, H k S k B · W (x0) = B · H · x0 = X, so R · x0 is contained in B · H · x0. Therefore, b∈B bAx0 = R . This implies that Ax0 must be cocompact so it must contain some element transverse to H.

We shall see the following case of Ax0 is important in the proof. r0 k ∼ Lemma 10.13. If Ax0 = H ⊕ Z` for some ` 6∈ H, then there is a C factor π : X → R /Ax0 = T k −1 0 of the R -action such that Hx0 = π (0), where r = (1, θ) or ∞, depending on whether the action is C2 or C∞, respectively.

k Proof. Fix y ∈ X, and let By ⊂ R be the set of points a such that (a + H)x0 3 y. We claim that By is a coset of Ax0 . Suppose that a1, a2 ∈ By, then y ∈ (a1 + H)x0 ∩ (a2 + H)x0. Then −1 a1 y ∈ Hx0 ∩ ((a2 − a1) + H)x0. As in the proof of Lemma 10.12, we may build a path from x0 −1 H to a1 y by choosing a long H-orbit segment, then a short path whose legs lie in W with a fixed k number of switches and a short α, −α, R -path. We may find a similar path from (a2 − a1)x0 to y. Concatenating these paths gives a path from x0 to (a2 − a1)x0. Since the number of switches is k uniformly bounded, we may push the α, −α, R -legs to the end of the path keeping them very short H using control on the number of switches. This shows that (a2 − a1)x0 ∈ W (x0) = Hx0, so that a2 − a1 ∈ Ax0 . Hence, By is a coset. k ∼ Now define π : X → R /Ax0 = T by π(y) = By. It is clear that the map determines a factor k −1 of the R -action, and by construction, Hx0 = π (0). We claim that π is continuous. Indeed, it −1 suffices to show that the preimage of a closed set is closed. Let C ⊂ T be closed. Then π (C) is homeomorphic to C × π−1(0) which is compact and hence closed. The homeomorphism is exactly given by (c, x) 7→ c · x. k k Choose a regular element a ∈ R . Since π is continuous and determines a factor of the R action, s u s it is constant on the leaves of Wa and Wa . Then π is constant on the leaves of Wa and smooth along k cs the R -orbits (since it is a factor), so by Journé’s theorem, π is smooth along the leaves of Wa . u Notice that it is also constant along Wa , so again applying Journé’s Theorem to the complementary cs u ∞ foliations Wa and Wa , we get that π is C globally. Proposition 10.14. If there is a periodic orbit p for which Case (2) of the dichotomy described at the start of the section holds, Theorem 2.1 holds.

k Proof. Ax0 contains H and H has codimension one, so either Ax0 = R or Ax0 = H ⊕ Z`. Since k k x0 has a dense R -orbit, if Ax0 = R , H · x0 = X, we arrive at the last conclusion of Theorem 2.1. Lemma 10.13 shows that if Ax0 = H ⊕ Z`, we have the second conclusion of Theorem 2.1. 60 11. Proofs of Theorems 2.1 and 2.2 First, we note that Theorem 2.1 follows immediately from Propositions 9.11 and 10.14 and the return dichotomy. In this section, we use Theorem 2.1 to prove Theorem 2.2. Proof of Theorem 2.2. If there is a non-Kronecker rank one factor the theorem holds, so assume that we either have the second or last conclusion of Theorem 2.1. By Lemma 3.13, we may without k k ` loss of generality assume that we have an R action (rather than an R × Z action). When H has a dense orbit, we may use Proposition 5.1 and the Livsic argument, to obtain the desired metric (as was done in [42], see Section 5.6). k Next, suppose we have a circle factor π : X → T of R y X, and that there exists x ∈ X0 = −1 α π (0) such that H · x is dense in X0. Define a metric on Eh·x by pushing forward any metric on α Ex . Then using Proposition 5.1, one may again show that the metric defined on the H-orbit of x extends continuously to an H-invariant metric on X0. This argument also shows the metric is k unique up to muliplicative constant. Choose any a ∈ R such that π(a · x) = 0, but π(ta · x) 6= 0 for t ∈ [0, 1). Then pushing the metric forward by a yields another H-invariant metric on X0, which must be a scalar multiple of the original one, so that ||a∗v|| = λ ||v|| for some λ > 0 and every α v ∈ E based at some point in X0. Notice that a 6∈ H since in this case π(ta · x) ≡ 0. Therefore, k k R = H ⊕ Ra. So we may define a functional α : R → R by α(h, t) = tλ. Finally, we define a metric on X in the following way: choose x ∈ X, so that x = ta·y for some y ∈ X0 and t ∈ R. Then α t −1 if v ∈ Ex , define ||v|| = λ (ta)∗ v . One may easily check that this metric and the functional α are well-defined and satisfy the conclusions of Theorem 2.2. Uniqueness follows immediately from the construction.

Part II. Constructing a Homogeneous Structure k r Throughout Part II, we assume that R y X is a C totally Cartan action, and that no finite k ` cover of the action has a non-Kronecker rank one factor (we reduce rigidity of R × Z actions to k+` rigidity of R action case in Section 16 by passing to a suspension). We can always pass to a finite cover of X on which the coarse Lyapunov foliations are oriented. By Theorem 2.1 and 2.2, we may and therefore do assume the following throughout Part II: Higher Rank Assumptions. β(a) a) Each functional β ∈ ∆ is chosen together with metrics ||·||β such that ||a∗v||β = e ||v||β k β for every a ∈ R and v ∈ W . The functional β is unique, and the ||·||β is unique up to global scalar. b) ker β either has a dense orbit, or has orbits dense in every fiber of some circle factor of the k R -action. Remark II.1. The second case of b) is mysterious, but we will be still able to prove that systems of this form are algebraic. However, we do not know if any such examples exist, we expect that they do not. Their existence is related to whether, on a nilmanifold, having no rank one factors implies the same feature for the actions on its center. It will be an annoyance, but not a real obstruction throughout the proofs in this part. Remark II.2. Throughout this section, each functional β is now defined uniquely, whereas in Part I (and again in Part III), it is only defined up to multiplicative constant. Therefore, unlike in those sections, we will take special care in considering the coefficients of the weights (so rather than saying the case of a pair of weights α and −α, we will need to consider the case of the pair α and −cα). Compare with Remark 3.9. 61 We also note that a) implies that the Lyapunov exponent for Eβ with respect to any invariant measure is equal to α (and not just some scalar multiple of α).

12. Basic Structures of Geometric Brackets First we define a fairly general notion, that of a topological Cartan action. Theorem 1.7 will be a corollary of the classification of such actions when they also have local product structure and certain resonance assumptions, see Theorem 12.19. The local product structure and resonance assumptions will always hold for C2 actions. We introduce this notion for three reasons: One of the main steps of the proof of Theorem 1.7 is to define a factor of such an action with much better transitivity properties for the coarse Lyapunov foliations. A priori, this factor is a compact metric space that may fail to be a manifold, and hence the action may not be smooth. While such a factor may not be a topological Cartan action at first, it inherits enough good properties of the action to produce a homogeneous structure on it. We then slowly show that all spaces and actions are indeed homogeneous. Second, it illustrates the versatility of the method for working with actions that may a priori fail to have any smoothness properties. Indeed, even for C∞ actions, the coarse Lyapunov foliations transversely are only Hölder, so it is not clear how to take Lie brackets of vector fields tangent to these foliations. We therefore replace standard tools of differential geometry with “geometric brackets”. These are motivated by the usual geometric interpretation of the Lie bracket but explicitly use the rigid structure of the coarse Lyapunov foliations coming from a higher-rank action. Third, it clarifies and emphasizes which structures of the smooth Cartan actions we will be using and is a useful reference for definitions in subsequent arguments. We begin by defining flows ηχ along the foliations W χ which will be critical to our analysis: χ χ Definition 12.1. For each χ ∈ ∆, let ηt denote the positively oriented translation flow along W (with respect to the norm ||·||χ), which satisfies

(12.1) a · ηχ(x) = ηχ (a · x) t eχ(a)t The next definition imitates the structure one naturally gets from a smooth Cartan action. In particular, we get one dimensional laminations generalizing the coarse Lyapunov spaces and parametrizations rescaled by the Cartan actions from Theorem 5.26. Properties (4), (5) and (6) in the definition are stronger and stronger transversality and accessibility properties. (4) is crucial to define our geometric brackets below. If Y is a metric space, let BY (y, ε) denote the ball of radius ε along y. k Definition 12.2. A transitive, locally free action of R on a compact finite-dimensional metric k ∗ space (X, d) is said to be a topological Cartan action if there is a set ∆ ⊂ (R ) (called the weights of the action) and a collection of locally free Hölder flows {ηχ : χ ∈ ∆}\{0} satisfying the following properties: (1) For any β ∈ ∆, cβ 6∈ ∆ for any c > 0, c 6= 1. χ (2) For every x ∈ X, t 7→ ηt (x) is locally bi-Lipschitz. (3) For every a ∈ k, t ∈ and x ∈ X, a · ηχ(x) = ηχ (a · x). R R t eχ(a)t k (4) If a1, . . . , am ∈ R is a list of Anosov elements, and {χ1, . . . , χr} is a circular ordering of − ¯ ∆ ({ai}), then for any combinatorial pattern β whose letters are all from Φ, η˜(Cβ¯)x ⊂ η˜(C )x =: W s (x) for every x ∈ X (recall Definition 4.2), and W s is the image of (χ1,...,χr) (ai) (ai) r r a continuous map from R to X, which on compact neighborhoods in R is a homeomorphism onto its image. 62 (5) If ηˆ is the corresponding action of Pˆ (see Definition 4.6), then ηˆ is transitive in the sense that for every x, y ∈ X, there exists ρ ∈ Pˆ such that ηˆ(ρ)x = y. s r (6) There exists a δ, ε > 0 such that if a is regular, y ∈ Wa (x, δ) :=η ˜(C(χ1,...,χr) ∩BR (0, δ)) and cu s cu z ∈ Wa (x, δ) := BRk (0, δ) · W−a(x, δ), then there is a unique intersection point Wa (y, ε) ∩ s Wa (z, ε). We say that the topological Cartan action has locally transverse laminations if it also satisfies

(7) for any ordering β¯ = (β1, . . . , βn) of ∆ which lists every weight exactly once, there exists k U ⊂ Cβ × R open containing 0 (the trivial path) such that for every x ∈ X, the restriction of the evaluation map at x from U → X is onto a neighborhood of x. Remark 12.3. Several of the properties of Cartan actions can be deduced from these axioms, including the Anosov closing lemma (Theorem 5.8) and density of periodic orbits (Theorem 5.13). We will use them freely for topological Cartan actions. Notice also that the higher rank assumptions still make sense, after replacing a) by (3). See Remark 5.6. k ` Remark 12.4. One may introduce the notation of a topological Cartan action of R ×Z in a similar k+` way, and prove similar results about them. They are related to R -actions by suspensions. Locally transverse laminations is weaker than the usual notion of product structure since we do not insist that the evaluation map is injective. On the other hand, it is slightly stronger than local transitivity. Recall that a family of foliations F1,..., Fm on a smooth manifold Y is locally transitive if for every ε > 0 there exists δ > 0 such for every y, z ∈ Y such that d(y, z) < δ, there exists Pk y = y0, y1, . . . , yk = z such that yi+1 ∈ Fi(yi) for some i ∈ {1, . . . , m}, and i=1 dFi(yi)(yi, yi+1) < ε. This is slightly weaker than locally transverse laminations, since each foliation can appear more than once. First let us explain how smooth Cartan actions relate to these topological notions.

Lemma 12.5. Let Y be a smooth manifold, and F1,..., Fm be uniquely integrable continuous Lm foliations with smooth leaves such that TyY = i=1 TyFi for every y ∈ Y . Then the foliations Fi have locally transverse laminations. Lemma 12.5 can be deduced in two ways: one may use a standard trick of approximating the vector fields tangent to the foliations Fi by smooth ones, obtaining locally transverse laminations using the inverse function theorem, and showing persistence of the transitivity using a degree argument. Alternatively, this is a more general phenomenon which does not require unique integrability, see [68, Lemma 3.2]. k 1,θ Proposition 12.6. Let R y X be a C transitive, totally Cartan action on a compact manifold without a rank one factor. Then if every coarse Lyapunov foliation is orientable, this data defines a topological Cartan action with locally transverse laminations. As discussed at the start of this section, we may always assume that the coarse Lyapunov foliations are orientable by passing to a finite cover. Proof. The special Hölder metrics from Theorem 5.26 determine unit vector fields, positively oriented, and thus flows which satisfy properties (1)-(3). We now show (4). Indeed, by Lemma 3.4, W s foliates the space and by Lemma 5.33, we have {ai} the corresponding parameterizations. (6) is immediate from Lemma 12.5. (5) follows from (6) and connectedness of X. k Definition 12.7. Fix a topological Cartan action R y X. Given a subset Φ = {β1, . . . , βn} ⊂ ∆, ∗|Φ| let η˜Φ denote the induced action of PΦ = R on X, and CΦ(x) ⊂ C(x) denote the stabilizer of x 63 ˆ k for η˜Φ, and C(x) = C∆(x). Let PΦ = R n PΦ, where the semidirect product structure is given as in k Pˆ = R n P (Definition 4.6).

Since our endgoal is understanding the structure of StabPˆ(x) for some x, the following lemmas provide important structures for paths coming from symplectic pairs.

k Lemma 12.8. Let R y X is a topological Cartan action, and suppose that α, −cα ∈ ∆ are negatively proportional weights. Then if x0 has a dense ker α-orbit (one of the cases of b)), C{α,−cα}(x0) ⊂ C{α,−cα}(x) for all x ∈ X and P{α,−cα}/C{α,−cα}(x0) is a Lie group. ˆ k −1 Proof. Notice that in the group P{α,−cα}, if a ∈ R , aC{α,−cα}(x)a = C{α,−cα}(a · x), and if −1 a ∈ ker α, aC{α,−cα}(x)a = C{α,−cα}(x) by (12.1). In particular, C{α,−cα}(·) is constant on ker α orbits. If ker α has a dense orbit, C{α,−cα}(x0) is contained in any C{α,−cα}(x) for every x. Hence, the quotient P{α,−cα}/C{α,−cα}(x0) is a Lie group by Corollary 4.13. Remark 12.9. Lemma 12.8 addresses the case of ker α having a dense orbit. If ker α only has orbits dense in the ﬁber of some circle factor, see Lemma 12.12. 0 1 0 0 Deﬁnition 12.10. Let g = sl(2, ). We call and the standard unipotent genera- R 0 0 1 0 1 0 tors, and the corresponding neutral element. Let h = Lie(Heis), where Heis is the standard 0 −1 0 x z  0 1 0   3-dimensional Heiseberg group, so that h = 0 0 y : x, y, z ∈ R . We call 0 0 0 and  0 0 0  0 0 0 0 0 0 0 0 1 0 0 1 the standard unipotent generators and 0 0 0 the corresponding neutral element. 0 0 0 0 0 0

Notice that in both sl(2, R), and h, the map which multiplies one of the standard unipotent generators by λ, the other by λ−1 and ﬁxes the corresponding neutral element is an automorphism of the Lie algebra. Let G{α,−cα} be the group P{α,−cα}/C{α,−cα}. 2 Proposition 12.11. If χ, −cχ ∈ ∆, then P{χ,−cχ}/C{χ,−cχ} is either isomorphic to R , (some cover χ −cχ of) PSL(2, R) or the Heisenberg group. Furthermore, in the last two cases, the ﬂows η and η are the one-parameter subgroups of the standard unipotent generators and the corresponding neutral element generates a one-parameter subgroup of the topological Cartan action.

Proof. G{χ,−cχ} is a Lie group by Lemma 12.8, which is generated by two one-parameter subgroups χ −cχ corresponding to the flows η and η . Let v± denote the elements of Lie(G{χ,−cχ}) which generate ηχ and η−cχ, respectively, and assume they don’t commute (if they commute we are done). By k Proposition 4.7, the action of a ∈ R intertwines the action of G{χ,−cχ} by automorphisms, and (1−c)χ(a) v0 = [v+, v−] is an eigenspace with eigenvalue e . By Lemma 4.10, the action of the one- parameter subgroup generated by v0 is Hölder, so the distance in the Lie group is distorted in only k a Hölder way. So for each element a ∈ R , if c > 1 and χ(a) > 0, each point in the orbit of the one parameter subgroup generated by v0 will diverge at an exponential rate. Similarly, one may see exponential expansion or decay properties for different choices of a and different possibilities for c. Therefore, if c 6= 1, the orbit of this one-parameter subgroup is contained in the coarse Lyapunov submanifold for either χ or −cχ. Since Cartan implies that each coarse Lyapunov foliation is one- dimensional, c = 1 and v0 is neither expanded nor contracted. Similarly, since [v0, v±] must be 64 ±χ(a) an eigenvalue of the automorphism corresponding to a with eigenvalue e , [v0, v±] ⊂ Rv±. In particular, as a vector space Lie(G{χ,−cχ}) is generated by {v−, v0, v+}. One quickly sees that the only possibilities for the 3-dimensional Lie algebras are the ones listed. χ k Assume c = 1 and let f denote the flow generated by v0. We claim there exists b ∈ R such χ χ that ft (x) = (tb) · x. We first show this for a sufficiently small, fixed t. Therefore, let f = ft and t be small enough so that d(f(x), x) < δ for every x ∈ X. Notice that f(a · x) = a · f(x), since by Proposition 4.7, v0 is an eigenvector of eigenvalue 0 for the resulting automorphism of G{χ,−χ}. k (α1) (αm) Fix a regular element a1 ∈ R . For each x, there exists a path of the form ρx = t1 ∗ · · · ∗ tm ∗ (β1) (βl) s1 ∗ · · · ∗ sl ∗ b such that f(x) = ρx ∗ x, and α1, . . . , αm, β1, . . . , βl is a circular ordering on the negative weights and positive weights, respectively (see Definition 5.30). We claim that ti = 0 for every i. Indeed, by applying a1 to ρ, we know that the αi components will all expand, and the b and sj components remain small. Thus, if δ is sufficiently small (relative to the injectivity radius of X), we may conclude that d(f(x), x) > δ for some x ∈ X, a contradiction. A similar argument shows that sj = 0 for every j. k Therefore, f(x) = bx · x for some bx ∈ R . Notice that if StabRk (x) = {0}, bx is unique and k continuously varying. Since f commutes with the R action, f(a · x) = ba·xa · x = abx · x, thus bx is k constant on free orbits. Since there is a dense R orbit, bx is constant, and f(x) = bx for all x ∈ X, as claimed. χ k χ Thus we have shown that each ft has an associated b(t) ∈ R such that ft (x) = b(t) · x for sufficiently small t. It is easy to see that b(t) varies continuously with t from the construction k χ above, and that the map t 7→ b(t) is a homomorphism from R to R . Hence, ft (x) = (tb) · x for k some unique b ∈ R . k Lemma 12.12. Let R y X be a Cartan action satisfying the higher rank assumptions, and suppose that α, −cα ∈ ∆ are negatively proportional weights. Then the action of Pα,−cα factors through the 2 action of a Lie group isomorphic to R , some finite cover of SL(2, R) or Heis. Proof. We combine the techniques of the previous arguments. Assumption b) of the higher rank assumptions implies that either ker α has a dense orbit (in which case we allude to Lemma 12.8), or ker α has a dense orbit on a fiber of a circle factor. In the latter case, it must have a dense orbit k on every fiber, since R can take one fiber to any other, and the action intertwines the action of ker α. Therefore, we may use the arguments of Lemma 12.12 and Proposition 12.11 to get that each 2 fiber has an action of R , Heis or a cover of SL(2, R), but that the group and action may, a priori, depend on the basepoint. Now, simply notice again that Cα,−cα(ax) = ψa(Cα,−cα(x)) by (12.1) and definition of ψa (see 2 Definition 4.6) for every a. But since ψa descends to automorphisms of R , Heis and any cover of SL(2, R), we conclude that ψa(Cα,−cα(x)) = Cα,−cα(x), and therefore the result. We can now establish the main result in this section, the principal technical tool in our analysis of k topological Cartan actions: the geometric commutator. Fix α, β ∈ ∆, and pick some regular a ∈ R such that D(α, β) ⊂ ∆−(a). Recall that D(α, β) comes equipped with a canonical circular ordering {α, γ1, . . . , γn, β} (see Definition 5.31). We use the following convention for group commutators: if G is a group and g, h ∈ G, then

(12.2) [g, h] = h−1g−1hg.

α,β Lemma 12.13. If t, s ∈ R, α, β ∈ ∆ are non-proportional, there exists a unique ρ : R×R×X → PD(α,β)\{α,β} such that: 65 (12.3) ρα,β(s, t, x) ∗ [s(α), t(β)] ∈ C(x). α,β For each χi ∈ D(α, β) \{α, β}, there is a unique function ρχi : R × R × X → R such that:

α,β α,β (χn) α,β (χ1) ρ (s, t, x) = ρχn (s, t, x) ∗ · · · ∗ ρχ1 (s, t, x) . α,β α,β Furthermore, ρ and ρχi are continuous functions in t, s, x, and

χi(a) α,β α,β α(a) β(a) (12.4) e ρχi (s, t, x) = ρχi (e s, e t, a · x).

ker β α

ker χ3 β

ker χ2

χ1 β

kerχ 1

χ2

χ3 ker α α

Figure 2. Canonical Commutator Relations

Proof. Since α and β are not proportional, if follows from (4) in the deﬁnition of a topological (β) (χn) (χ1) (α) Cartan action that there exists some unique path ρx = u2 ∗ vn ∗ · · · ∗ v1 ∗ u1 ∈ PD(α,β) (β) (α) satisfying ρx ∗ [t , s ] ∈ C(x). We wish to show that ρx has trivial α and β components. Notice that by linear independence of α, β, we may choose a such that α(a) = 0 and β(a) < 0. Then notice that by (12.1), for any x ∈ X:

h i h i (12.5) (na) s(α), t(β) (−na) · x = s(α), enβ(a)t(β) x −−−→n→∞ x.

Since the action of η˜ is Hölder, the convergence is exponential since β(a) < 0. But if ρx has a nontrivial α component, that component does not decay, since it is isometric. Therefore it must be trivial. By a completely symmetric argument, the β component is trivial. Deﬁnition 12.14. If α, β ∈ ∆ are linearly independent weights, let n α,β o [α, β] = χ ∈ ∆ : ρχ (s, t, x) 6= 0 for some t, s ∈ R, x ∈ X . It is clear that [α, β] = [β, α], since ρβ,α(t, s, ρα,β(s, t, x) · x) = ρα,β(s, t, x)−1. Notice also that, α,β a priori, each ρχ may vanish at some x ∈ X, but not every x. We will see later that the no α,β rank-one factor assumption implies that ρχ is actually a polynomial independent of x. 66 12.1. Further Technical Tools. In the remainder of this section we establish some further technical results related to the geometric commutators which will be useful in future sections.

− k Lemma 12.15. Let Ω ⊂ ∆ (a) for some Anosov a ∈ R . Suppose that the action of PΩ factors through a Lie group action H. Then (1) H is nilpotent, (2) if α, β ∈ Ω and χ ∈ [α, β], then χ = uα + vβ with u, v ∈ Z+, and α,β u v (3) if α, β ∈ Ω, then ρuα+vβ(s, t, x) = cs t for some c ∈ R.

Proof. By Proposition 4.7, the automorphism ψa descends to an automorphism of H. Since the eigenvalues dψ¯ are all less than 1, ψ¯a is a contracting automorphism of H, and H is nilpotent. If α,β ρχ (s, t, x) 6≡ 0, then the subgroups of H corresponding to α and β do not commute. Further- more, the Baker-Campbell-Hausdorff formula, together with the end of Proposition 4.7(3), implies that {uα + vβ : u, v ≥ 0, u, v ∈ Z} is a closed subalgebra. In particular, if a weight χ satisfies α,β ρχ (s, t, x) 6≡ 0, then χ must have integral coefficients, as claimed. Finally, the Baker-Campbell- α,β Hausdorff formula also implies that each ρχ is a polynomial as H is nilpotent, and (12.4) implies that this polynomial is u-homogeneous in s and v-homogeneous in t. Much of this section has been devoted to establishing properties of totally Cartan actions from smooth ones. In fact, a C1,θ totally Cartan action enjoys all of the properties discussed. We return once again to the smooth setting to establish one more property. Unfortunately, we will require slightly higher regularity: a C2 action. Indeed, in the proof of the next lemma, we use that the regularity of the holonomies along the fast foliation of a C2 diffeomorphism is Lipschitz. Whether this property holds for C1,θ diffeomorphisms is open.

k 2 Lemma 12.16. If R y X is a C totally Cartan action on a smooth manifold with no non- Kronecker rank one factors, α, β ∈ ∆ are linearly independent, and χ = uα + vβ with u < 1 and v < 1, then χ 6∈ [α, β]. Proof. Divide D(α, β) = A ∪ B, where A = {uα + vβ : u ≥ 1} ∩ ∆ and B = {uα + vβ : u < 1} ∩ ∆. Let B0 = {uα + vβ : u < 1, v ≥ 1} ∩ ∆. Give the weights of D(α, β) the canonical circular ordering 0 0 {α = χ1, . . . , χn = β}. We will show that if χi, χj ∈ A ∪ B , then [χi, χj] ⊂ A ∪ B by induction on |i − j|. Then the result follows since α = χ1 and β = χn. The base case is trivial: if |i − j| = 1, then χi and χj are adjacent in the circular ordering, and 0 therefore commute. We now try to commute χi, χj ∈ A ∪ B , |i − j| > 1. Notice that choosing a ∈ ker β, and perturbing by a very small amount will yield an element a0 close to a for which L χ L B χ∈A T W ⊕ χ∈B T W is a dominated splitting (cf. slow foliations, Section 8.2). Therefore, L χ A A 0 χ∈A T W is tangent to a foliation W . Notice that each leaf W has local C charts given by χ motion along each W , χ ∈ A (the proof goes as in Lemma 5.33). In particular, if both χi 0 and χj ∈ A, [χi, χj] ⊂ A. Similarly, if both χi and χj belong to B , then they both belong to C = {uα + vβ : v ≥ 1} ∩ ∆. So for similar reasons, choosing a perturbation of b ∈ ker α, gives [B0,B0] ⊂ [C,C] ⊂ C ⊂ A ∪ B0. 0 We now consider the case when χi ∈ B and χj ∈ A with |i − j| > 1 (the last case follows from a symmetric argument). Let x ∈ X, y = t(χi) · x, x0 = s(χj ) · x, y0 = s(χj ) · y and w = t(χi) · x0. Notice that y0 = [(−s)(χj ), (−t)(χi)] · w. Let Dij = {χi+1, . . . , χj−1} be the set of weights strictly between χi and χj. Decompose Dij into {γ1, . . . , γm1 , δ1, . . . , δm2 , 1, . . . , m3 }, where {γk}, {δk} and {k} are the weights of A ∩ Dij, 0 0 (B \ B ) ∩ Dij and B ∩ Dij, respectively, each listed with the induced circular ordering. We assume 67 y and w are sufficiently close, to be determined later (if we show it for sufficiently small s, t, we may k use the dynamics of R y X and (12.1) to conclude it for arbitrary s, t). Lj−1 χk ij 2 Notice that the distribution k=i+1 T W is uniquely integrable to a foliation W with C leaves since it is the intersection of stable manifolds for the action. Since y0 = [(−s)(χj ), (−t)(χi)]·w, y0 ∈ Wij(w) by Lemma 12.13. Therefore, by Lemma 12.5 applied to the splitting T Wij into the bundles Eγk , Eδk and Ek , there exists a path moving from w to y0 which first moves along the 0 leaves of the 1-dimensional foliations corresponding to B ∩ Dij, then the 1-dimensional foliations 0 corresponding to (B \ B ) ∩ Dij, then the 1-dimensional foliations corresponding to A ∩ Dij, in the given orderings. Let p denote the point obtained after moving along the B0 foliations from w, and q denote the point obtained by moving from p along the B \ B0 foliations. Then q is also connected to y0 via the A foliations. See Figure3.

w p 0 B B \ B 0 q 0 x t(χi) A

s(χj ) s(χj )

x t(χi) y

Figure 3. A geometric commutator

2 0 Choose any pair of C , |B|-dimensional embedded discs D1 3 x, y, D2 3 x , q transverse to A |α,β| 0 0 W inside of W (x). This is possible since x and y are connected via χi ∈ B and x and q are connected via only weights coming from B. Therefore, x0 and q are the images of x and A y under the W holonomy from D1 to D2. By [49, Section 8.3, Lemma 8.3.1], there exist bi- Lipschitz coordinates for which the leaves of the foliation WA are parallel Euclidean hyperspaces. A In particular, the holonomy along W is uniformly Lipschitz independent of the choice of D1 and 0 D2, given sufficiently good transversality conditions, so d(x, y)/d(x , q) is bounded above and below by a constant. 0 We claim that p = q (ie, that no weights of (B \ B ) ∩ Dij appear). Roughly, the reason is that such weights contract too slowly. Indeed, pick an element a0 ∈ ker α such that β(a0) = −1. We may perturb a0 to an element a which is regular and such that α(a) < 0, and such that if 0 0 δ ∈ B \ B , χi(a) < δ(a) < 0. This is possible because if δ = uα + vβ ∈ B \ B , then v < 1, so 0 0 0 0 δ(a ) = −v > −1 = β(a ) ≥ χi(a ), since when χi ∈ B , the β coefficient is at least 1. This is clearly 68 an open condition for each δ, so we may choose a as indicated. We may also assume, by choosing a multiple, that β(a) = −1. Since y ∈ W χi (x), we can estimate distance between iterates of x and y using the Hölder metric χ 0 along the coarse Lyapunov leaf W i (x). Recall that since χi ∈ B , χi(a) < β(a) = −1. Therefore k k kχi(a) −k dW χi (a · x, a · y) = e dW χi (x, y) < e dW χi (x, y) using the Hölder metric on the manifold. 0 Now, suppose p 6= q. Recall that p and q are connected by legs in B \ B = {δ1, . . . , δm}, so that δ δ there exist p = x0, x1, . . . , xm = q such that xl ∈ W l (xl−1). Since the foliations W l are transverse, if p 6= q, there exists some l for which xl 6= xl−1. Without loss of generality, we assume that {δl} are ordered such that 0 > δ1(a) > δ2(a) > ··· > δm(a) > χi(a). Then let l0 be the minimal l for which xl 6= xl−1, and c1 = δl0 (a), c2 = δl0+1(a). Notice that 0 > c1 > c2 > −1. By minimality, we get xl0−1 = p. Let d denote the Riemannian distance on the manifold. Since for any γ ∈ ∆, the distance along each W γ leaf is locally Lipschitz equivalent to the distance on the manifold, there exists L > 0 such γ 0 −1 0 0 that for all γ ∈ ∆ and sufficiently close points z ∈ W (z ), we have L dW γ (z, z ) ≤ d(z, z ) ≤ 0 LdW γ (z, z ). Then after applying the triangle inequality, we get:

k 0 k k k k 0 k k k d(a · x , a · q) ≥ d(a · xl0 , a · p) − d(a · x , a · p) − d(a · xl0 , a q) −1 k k k 0 k k k ≥ L d δ (a · x , a · p) − d(a · x , a · p) − d(a · x , a q) W l0 l0 l0 −2 c1k 2 c2k 0 c1k ≥ L e d(xl0 , p) − L Ce ≥ C e

since by construction, we may iteratively apply the triangle inequality to all legs connecting xl0 and 0 c k −k q and x and p, which contract faster than e 2 and e , respectively, since c1 > c2, −1. We may A k construct new disks D1,k and D2,k with suﬃcient transversality conditions to W connecting a · x k k 0 k and a · y, and a · x and a · q (note that we may not simply iterate the disks D1 and D2 forward, since the transversality may degenerate). Therefore, since each of the foliations along weights of B are uniformly transverse to those of A, and the holonomies are Lipschitz with a uniform Lipschitz constant on suﬃciently transverse discs, we arrive at a contradiction, so p = q. Therefore, the connection between w and y0 only involves weights of A ∪ B0. By the induction 0 hypothesis, the commutator of two weights χk, χ` ∈ (A ∪ B ) ∩ Dij produces only weights in (A ∪ 0 B ) ∩ Dk`. We may therefore reorder the weights appearing here using commutator relations (which a priori depend on the basepoint) to put the relations in a desired circular ordering without weights in B\B0. This proves the inductive step, and hence the lemma, since B\B0 = {uα + vβ : u, v < 1}∩ ∆.

Remark 12.17. Lemma 12.16 is the only place in which we use the fact that the action is C2, so that we may apply the regularity of holonomies. In fact, if one had that the holonomies were biLipschitz for C1,θ actions, then one could produce proofs of the main theorems for C1,θ actions. We say that a topological Cartan action has sub-resonant Lyapunov coeﬃcients if the conclusion of Lemma 12.16 holds.

1,θ k Given a C -embedded totally Cartan subaction of R y X, the holonomies used in the proof of Lemma 12.16 will coincide with the ones on X. This gives the following as an immediate corollary, which we will use in Part III:

k 1,θ Corollary 12.18. Suppose that R y X0 is a C , totally Cartan action without rank one factors 1,θ 2 k0 k that C -embeds into a C action R y X. Then R y X0 has subresonant Lyapunov coeﬃcients. 69 With subresonant Lyapunov coeﬃcients, one may prove Theorem 1.7 for topological Cartan actions, which we now formulate. For the rest of Part II, we will not rely on the smoothness properties, only the structures obtained in this section, so we aim to prove the following:

k Theorem 12.19. Let R y M be a topological Cartan action with locally transverse laminationss k and sub-resonant Lyapunov coefficients. Assume that ker α ⊂ R has a dense orbit for every α ∈ ∆. k Then there exists an homogeneous Cartan action R y G/Γ which is topologically conjugate to k 1,θ ∞ 1,θ ∞ R y M. If the topological Cartan action is C or C , then the conjugacy is C or C , respectively. We note that under additional ergodicity assumptions, C∞ totally Cartan actions for k ≥ 3 were k treated in [42], using transitivity of restrictions to subgroups of the form ker α ∩ ker β ⊂ R for 2 arbitrary α, β ∈ ∆. Their method completely fails for R Cartan actions. The stronger transitivity 2 assumption dooms the approach: in R , ker α ∩ ker β is an intersection of lines, and hence trivial. Thus, the main achievement of this part is establishing the rank 2 case and using only the transitivity condition provided by Theorem 2.1, and requires entirely new ideas and methods. These methods also allow us to work with C2 actions. Remark 12.20. The main result of [42] is a corollary of Theorem 1.7, but an improved version can be obtained more directly with our tools. In particular, we no longer require an invariant measure of full support. The proof is much simpler but requires transitivity of actions of codimension two hyperplanes. We treat this special case in Section 14.1, which does not rely on Section 13. The reader may therefore skip Section 13 and Sections 15-16 to obtain the special case. In the remainder of Part II, we will prove that certain canonical cycles are all constant, and that such constancy of such cycles are sufficient for homogeneity. For this we first prove that geometric brackets are constant.

13. Ideals of Weights Our main result has a much simpler proof if all coarse Lyapunov foliations have dense leaves. 2 However, this is not always the case, even for R actions. For example one may take a twisted Weyl chamber flow (Section 6.1.3). In such examples, the coarse Lyapunov foliations tangent to a fiber are at best dense in that fiber. This leads us to introduce natural quotients of Cartan actions obtained by collapsing whole coarse Lyapunov foliations. Such factors are closely tied with geometric brackets and we are naturally led to study “ideals” of weights (Definition 13.1). The main insight here is that the geometric commutators (Lemma 12.13) treat dynamical ideals in the same way that Lie brackets treat algebraic ideals. See Lemma 13.5. Definition 13.1. Given a subset E ⊂ ∆, let AE(x) be the set of all points y ∈ X which are endpoints of broken paths using the foliations from E starting at x, together with their accumulation points. Then set 0 χ I(E) = f ∈ C (X): f(ηt (x)) = f(x) for every x ∈ X, t ∈ R, χ ∈ E E T −1 and M (x) = f∈I(E) f (f(x)). The ideal closure of E is the set ¯ χ E E E = χ ∈ ∆ : ηt (M (x)) = M (x) for every x ∈ X, t ∈ R . We call E ⊂ ∆ an ideal if E¯ = E. An ideal E is called proper if E 6= ∅ and M E(x) 6= M. We say that E is maximal if it is not properly contained in any proper ideal. 70 Remark 13.2. One may alternatively define a proper ideal to be a subset E for which E 6= ∅ and k E 6= ∆. This definition turns out to be the same in most cases, but not all. If the R action is k the suspension of an Anosov action Z y X, then ∆ is a proper ideal according to our definition. k k Indeed, the factor we construct for this system will be the usual action R y T , and the fibers will be diffeomorphic to X, and foliated by the coarse Lyapunov foliations. The definition we make plays a crucial role in our arguments, eg, the proof of Lemma 13.4.

Remark 13.3. In the homogeneous setting, an ideal of weights will be a set of weights E such that L χ χ∈E g (together with some subalgebra of the acting group) is an ideal in g whose orbit in the homogeneous space is closed. Thus, an ideal of weights for a homogeneous action determines an ideal in g, but the opposite is not always true. For instance, one may take a homogeneous space (G×G)/Γ, where G is semisimple and Γ ⊂ G×G is an irreducible lattice. Then for the corresponding Weyl chamber flow, the Lie algebra of the first copy of G is an ideal in the Lie algebra sense, but the collection of weights from the first copy is not an ideal in our sense. Indeed, the corresponding weights generate the first copy, and this copy has dense orbits in the homogeneous space.

Notice that M E(x) are all closed sets which contain the sets AE(x).

Lemma 13.4. If E ⊂ ∆ is a maximal ideal, then for every weight χ ∈ ∆ \ E, the only continuous functions constant under ηχ and ηβ for every β ∈ E are constant.

Proof. Let χ ∈ ∆ \ E, and consider {χ} ∪ E. Since E is maximal, E0 = E ∪ {χ} satisﬁes M E0 (x) = X. Then if f is ηχ- and ηβ-invariant for every β ∈ E, it is invariant for all γ ∈ E0, and hence E0 constant on M (x) = X.

Lemma 13.5. If E is an ideal, α ∈ E and β ∈ ∆ are linearly independent, and χ ∈ [α, β], then χ ∈ E.

α,β Proof. We first claim the following: if χ ∈ [α, β], the set of points for which ρχ (s, t, x) 6= 0 for α,β some s, t ∈ R is open and dense. Indeed, openness follows from continuity of ρ (s, t, ·) for each k fixed s, t ∈ R. Then density follows from this set being R -invariant by (12.4), and our assumption k that R has a dense orbit. α,β Since the periodic orbits are dense by Theorem 5.13, and the set of points satisfying ρχ (s, t, ·) 6≡ 0 α,β is open and dense for every χ ∈ [α, β], we may fix some x ∈ X such that ρχ (s, t, x) 6≡ 0 for every χ ∈ [α, β]. α,β Choose s, t ∈ R and x ∈ X such that ρχ (s, t, x) 6= 0 (without loss of generality, we may assume k it is positive) and x is periodic. We first deal with the case [α, β] = {χ}. Choose a ∈ R such that α(a) > 0, β(a) = −1 and χ(a) = 0. Let f ∈ I(E), so that in particular, f is ηα invariant. Then k choose rk → +∞ such that (rka) · x → x (since x is on an R periodic orbit, this is possible). Then α,β if u = ρχ (s, t, x), 71 f(x) = lim f((rka) · x) k→∞ α = lim f(η α(a)r ((rka) · x)) k→∞ se k β α = lim f(η −r η α(a)r ((rka) · x)) k→∞ te k se k α β α = lim f(η α(a)r η −r η α(a)r ((rka) · x)) k→∞ −se k te k se k β α β α = lim f(η −r η α(a)r η −r η α(a)r ((rka) · x)) k→∞ −te k −se k te k se k α β = lim f((rka) · [ηs , ηt ]x) k→∞ χ = lim f((rka) · ηu (x)) k→∞ χ = lim f(ηu ((rka) · x)) k→∞ χ = f(ηu (x)).

Notice that the equality on the second equation and the fourth equation follows from the ηα invariance, and the third line and fifth equation follows from the uniform continuity of f and the −r α,β fact that e k → 0. Notice also that by sending s → 0, the corresponding ρχ (s, t, x) = u → 0. χ Therefore, we have f(x) = f(ηv (x)) for all v ∈ [0, u) and all periodic x ∈ X in a dense open set. Therefore it holds for all x ∈ X since periodic points are dense. Since it holds for every x and χ [0, u) generates R+, it holds for every u ∈ R+, and hence u ∈ R. Therefore, f(ηu (x)) = f(x) for all x ∈ X, so f is χ-invariant and χ ∈ E by definition of an ideal. Now we address the general case, in which ρα,β(s, t, x) is written as a product of legs in D(α, β) \ {α, β} = {γ1, . . . , γn}. Notice that we do not assume that γi ∈ [α, β], since we will need to take brackets of weights appearing in [α, β] later, and we do not yet know that they are closed under α,β commutators. For those which are not in [α, β], note that ργ (s, t, x) ≡ 0. We order the γi in the usual way so that there exists ai with α(ai) > 0, β(ai) < 0, γi(ai) = 0, and γj(ai) < 0 if j > i and γj(ai) > 0 if j < i. That is, γ1 is the closest to α in the circular ordering and γk is the closest to β. Recall that the ρ-functions are defined so that the endpoint of the commutator [s(α), t(β)] · x is α,β (γ1) α,β (γn) α,β (−ργ1 (s, t, x)) ∗ · · · ∗ (−ργn (s, t, x)) ·x. Let zi = −ργi (s, t, x) for notational convenience. Now, we adapt the argument from the case when [α, β] = {χ}: because we may choose a1 ∈ ker γ1 which contracts γi for all i > 1 and β, the argument is almost identical by choice of a1. Notice that if z1 ≡ 0 (for all s, t, x), γ1 6∈ [α, β]. If z1 6≡ 0, one may again get that z1 6= 0 on an open dense set k of points. In particular, f is γ1-invariant since this may be done at any R periodic point, and such points are dense. Now, we may repeat the argument, skipping each zi if zi ≡ 0, and using the induction hypothesis that f is γj invariant for all j < i for which zj 6≡ 0. The argument when [α, β] = {χ} still applies, now choosing ai ∈ ker γi. In this case, the only legs which are expanded are α and γj, j < i under which we know f is invariant. The formal proof goes as follows. Define

(γ ) (γn) γ (a) (γ ) γn(a) (γn) σi = (zi+1) i+1 ∗ · · · ∗ (zn) σi(a) = ψa(σi) = (e i+1 zi+1) i+1 ∗ · · · ∗ (e zn) (γ1) (γi−1) −1 γ1(a) (γ1) γi−1(a) (γi−1) −1 τi = (z1 ∗ · · · ∗ zi−1 ) τi(a) = ψa(τi) = ((e z1) ∗ · · · ∗ (e zi−1) ) 72 (α) (β) −1 (γi) By construction, we have chosen σi and τi so that [s , t ] · x = τi ∗ zi ∗ σi · x, and a similar equation holds after conjugating by a. Choose ai be such that γi(ai) = 0, γj(ai), β(ai) < 0 if j > i and γj(ai), α(ai) > 0 if j < i. In particular, σi(ra) tends to the trivial path if R 3 r → +∞. Choose real numbers rk → +∞ such that rkai · x → x, so that:

α(ai)rk (α) f(x) = lim f((rkai) · x) = lim f((se ) · (rkai) · x) k→∞ k→∞

β(ai)rk (β) α(ai)rk (α) = lim f((te ) ∗ (se ) · (rkai) · x) k→∞

α(ai)rk (α) β(ai)rk (β) α(ai)rk (α) = lim f((−se ) ∗ (te ) ∗ (se ) · (rkai) · x) k→∞

β(ai)rk (β) α(ai)rk (α) β(ai)rk (β) α(ai)rk (α) = lim f((−te ) ∗ (−se ) ∗ (te ) ∗ (se ) · (rkai) · x) k→∞ h i α(ai)rk (α) β(ai)rk (β) = lim f( (se ) , (te ) · (rkai) · x) k→∞ h i α(ai)rk (α) β(ai)rk (β) = lim f(τi(rkai) ∗ (se ) , (te ) · (rkai) · x) k→∞

(γi) = lim f(zi ∗ σi(rkai) · (rkai) · x) k→∞

(γi) (γi) = lim f(zi · (rkai) · x) = f(zi · x). k→∞ Proposition 13.6. The sets M E(x) can be obtained by the following transﬁnite induction: Let E E S0(x) = A (x), Σi+1(x) = {y ∈ M : Si(x) ∩ Si(y) 6= ∅} and Si+1(x) = Σi+1(x). Then M (x) = Sω(x) is the limit of this procedure for any ordinal ω such that Sω(x) = Sω+1(x) for every x ∈ X.

Proof. First notice that the ordinal ω may be chosen independent of x, since the sets Sω(x) cannot continue to grow in cardinality beyond the continuum. The limits of the Si’s are closed because they terminate when Si(x) = Si+1(x), so Si(x) = Σi+1(x). Furthermore, they partition the space, since on the terminating ω, if Sω(x) ∩ Sω(y) 6= ∅, then y ∈ Σω+1(x) ⊂ Sω+1(x) = Sω(x), by definition. E We claim that the quotient topology on the space of sets X = {Sω(x): x ∈ X} is a compact Hausdorff topology. Compactness follows generally, so we must show Hausdorff. If XE is not Hausdorff, then there exists Sω(x) 6= Sω(y) such that any pair of saturated neighborhoods U ⊃ Sω(x) and V ⊃ Sω(y) intersect at some point z(U, V ). Let d(x, S) = infy∈S d(x, y) be the usual distance between a point and a set. If S ⊂ X is a set, let S B(S, ε) = {y : d(y, S) < ε} = x∈X B(x, ε). Let Uk = B(Sω(x), 1/k) and Vk = B(Sω(y), 1/k), and zk ∈ X be in the intersection. Since zk ∈ X and X is compact, there is a convergent subsequence zkl → z ∈ X. Since Sω(x) and Sω(y) are both compact and d(zk,Sω(x)), d(zk,Sω(y)) < 1/k, we get that z ∈ Sω(x) ∩ Sω(y). By the transfinite induction, Sω(x) = Sω(y), a contradiction. E By construction, Sω(x) ⊂ M (x), since all continuous functions evaluate the same on each Si E by (transfinite) induction. Furthermore, the map M → X defined by x 7→ Sω(x) is a continuous function which separates the sets Sω(x). Since each Sω(x) is invariant under the flows from E, we get that we may separate points from different such Sω’s with E-invariant continuous functions into a compact Hausdorff space, and therefore E-invariant continuous functions into R by the Urysohn E lemma. Hence, M (x) ⊂ Sω(x). If E is an ideal, let XE = X/ ∼ be the set of equivalence classes of ∼, where x ∼ y if and only if M E(x) = M E(y) (as defined in the previous proof). 73 Recall that if α and −cα are both weights, then Proposition 12.11 implies that they generate a 2 group action of R , Heis or a cover of PSL(2, R). The next lemma is just a calculation but it is crucial here and again later in Lemma 14.11.

+ − Lemma 13.7. Let H cover PSL(2, R) and denote by hs and hs the unipotent flows in H covering 1 s 1 0 t 0 and respectively. Also let at = 1 . 0 1 t 1 0 t If st 6= −1 then + − − + hs ht = h t hs(1+st)a1+stz(s, t) 1+st where z(s, t) belongs to the center of H. If s and t are sufficiently small, z(s, t) = e. Proof. If st 6= −1 then 1 s 1 0 1 0 1 s(1 + st) 1 + st 0 = 0 1 t 1 t/(1 + st) 1 0 1 0 1/(1 + st) Therefore the claim holds in PSL(2, R), and thus in H up to some central element z(s, t). If s and t are sufficiently small, all calculations take place in a neighborhood of e, so the central element z(s, t) = e. Lemma 13.8. Fix an ideal E and a weight β ∈ ∆. β E E β (1) If −cβ 6∈ E for any c > 0, ηt (M (x)) = M (ηt (x)). β E E β (2) If β = −cα for some α ∈ E and β commutes with α, ηt (M (x)) = M (ηt (x)). (3) If β = −α for some α ∈ E and together they generate Heis, then Z(Heis) · M E(x) = M E(x) β E E β and ηt (M (x)) = M (ηt (x)). (4) If β = −α for some α ∈ E and together they generate a cover of PSL(2, R) with α, then E β E E β β ∈ ∆, the cover of PSL(2, R) leaves M (x) invariant, and ηt (M (x)) = M (ηt (x)). E β E β Proof. We first prove (1). We wish to show that if y ∈ M (x), then ηt (y) ∈ M (ηt (x)). If y is reached from a broken path along the foliations of E, this follows from Lemma 13.5. Since the actions ηβ are continuous, this passes to the closure, AE(x) = S0(x) as defined in Proposition 13.6. Inductively and using continuity, we get that it is true for every Si(x), in particular for E β E β Sω(x) = M (x). So ηt (y) ∈ M (ηt (x)). The proof of (2) is identical to that of (1), with the additional observation that we may replace our use of Lemma 13.5 when commuting β with α with the assumption they commute, if y must be reached using a path containing an α leg. For (3), we first show that Z(Heis) · M E(x) = M E(x). The proof is similar to that of Lemma k α −α 13.5. If f is an α-invariant function and x ∈ X is R -periodic, consider y = [ηt , ηt ] · x = z · x k for some z ∈ Z(Heis). Applying some a ∈ R for which α(a) = 1, we may choose rk → ∞ such that rka · x → x. Then in the limit, the α legs of the commutator become arbitrarily long, which does not affect the value of f since f is ηα-invariant. On the other hand, the −α legs become arbitrarily small, so difference between f(x) and f(y) can be made arbitrarily small. Therefore, f(x) = f(y) = f(z · x). Since periodic points are dense, we get this for every x. By choosing different values of t ∈ R, we may generate the entire center, giving the first part of (3). For the second part of (3), notice that if y is reached by a broken path whose weights come from E, we may again commute β = −α with weights in E which are not α using Lemma 13.5. When commuting β with α, we obtain another point of M E(x) by the first part of (3). Thus we get the claim for every y reached by a broken path whose weights come from E, and again by transfinite induction and continuity we get the full claim. 74 k Finally, we come to (4). Let {au : u ∈ R} be the 1-parameter subgroup of R corresponding to the diagonal group of the cover of PSL(2, R). Notice that for sufficiently small u, Lemma 13.7 (α) (−α) (α) (−α) k allows us to write au · x as t1 ∗ s1 ∗ t2 ∗ s2 · x. We may apply an element b ∈ R such that α α(b) = 1 and if x is periodic we may choose rk → ∞ such that rkb· x → x. Then for an η -invariant function f, we conclude that f(au · x) = f(x) in the same way as in (3), and extend to every x ∈ X by density of periodic orbits. Since f is invariant under au, it is also constant along any foliation β uniformly expanded or contracted by au. In particular, f is invariant under η and since E is an E β ideal, β ∈ E. Since β ∈ E, the sets M (x) are invariant under η , and we are done. Remark 13.9. It is tempting to believe that when α and −α generate a copy of Heis, α ∈ E implies that −α ∈ E, which is the case for covers of PSL(2, R). However, this is not true, as illustrated at the end of Section 6.3.1. The following is immediate from Lemma 13.8 and (12.1). E Proposition 13.10. The Cartan action and flows ηβ with β 6∈ E descend to actions on X . k E We call the induced action R y X an ideal factor of the Cartan action on X. We first upgrade the topological structure on XE by endowing it with a canonical metric, the Hausdorff metric. Corollary 13.11. XE is a compact metric space when given the Hausdorff metric. Proof. Since the points of XE are compact sets, we only need to show that XE is a closed subset of all compact subsets of X. We use the holonomies of the orbit of the Cartan action and the coarse χ E Lyapunov flows η . Suppose that M (xk) converges to some compact K ⊂ X, and note that we E may choose xk → x for some x ∈ K. We first show M (x) ⊂ K. It suffices to show that any χ E function invariant by η , χ ∈ E is constant on K. Any such function is constant on M (xk), so it is constant on the limit set, K. We now show the opposite inclusion. Since xl → x, xl is eventually in a small enough neighborhood for which the map (a, t , . . . , t ) 7→ a · ηχ1 . . . ηχn (x) from k × n to X contains x in 1 n t1 tn R R l the image by locally transverse laminations (Condition (6) of Definition 12.2). Hence there exists k n (al, tl) ∈ R × R converging to 0 such that xl = al · ηtl (x). Then let Fl be the restriction of −1 E E (al · ηtl ) to M (xl), which takes values in M (x) by Proposition 13.10. Notice that since the sequence of maps Fl are restrictions of transformations converging to the identity on X, for every E yl ∈ M (xl), d(yl,Fl(yl)) → 0 as ` → 0, independently of the choice of yl. Let y ∈ K, so that there E E exist yl ∈ M (xl) such that yl → y. Let zl = Fl(yl) ∈ M (x). By construction, zl → y, so since E E E M (x) is compact, y ∈ M (x). That is, K ⊂ M (x). While Proposition 13.10 shows that the flows descend to the quotient spaces XE, it is not obvious that the quotient actions are topological Cartan actions as defined in Definition 12.2. For instance, it is not clear that the flows ηχ, χ 6∈ E are locally free (even though they are faithful). Furthermore, the space XE is not guaranteed to be finite-dimensional, even though X is. However, one may attempt to define continuous functions ρα,β(s, t, x) by considering the definitions in the original α,β space X, and ignoring any terms ρχ (s, t, x) with χ ∈ E. Since the relation (12.3) holds on X, it must also hold for the induced flows on XE, provided it is uniquely defined. Thus, for an ideal factor, α,β the following lemma provides a canonical definition for ρχ , when α, β, χ 6∈ E. If α, β ∈ ∆ \ E, let DE(α, β) = D(α, β) \ E.

Lemma 13.12. If E is an ideal, α, β ∈ ∆ \ E are linearly independent, and χ ∈ DE(α, β), the α,β γ functions ρχ (s, t, x) are invariant under the flows ηt for every γ ∈ E (and therefore descend to functions on XE). 75 Proof. Recall that ρα,β(s, t, x) is defined to be the unique path in the legs D(α, β) \{α, β} with a fixed circular ordering such that ρα,β(s, t, x) ∗ [s(α), t(β)] ∈ C(x). In particular,

(13.1) ρα,β(s, t, x) ∗ (−t)(β) ∗ (−s)(α) ∗ t(β) ∗ s(α) · x = x (13.2) (−s)(α) ∗ t(β) ∗ s(α) · x = t(β) ∗ ρα,β(s, t, x)−1 · x (13.3) (−t)(β) ∗ (−s)(α) ∗ t(β) · y = ρα,β(s, t, x)−1 ∗ (−s)(α) · y, where y = s(α) · x. In particular, given any conjugation of linearly independent weights, the action coincides with the action of the conjugated element, times some ρ-function which can appear on the left or right at our preference. The idea is to apply a γ holonomy to this cycle, which is equivalent to the above conjugation action. Observe that by Lemma 13.5, as γ ∈ E, the new terms appearing are in E, so if α, β, χ 6∈ E, then the values remain the same. Formally the argument goes as follows: consider the path u(γ) ∗ ρα,β(s, t, x) ∗ [s(α), t(β)] ∗ (−u)(γ). This path fixes u(γ) · x since the middle of the conjugation α,β Q1 α,β α,β fixes x. Recall that by definition, ρ (s, t, x) = i=n ρχi (s, t, x). Expanding ρ (s, t, x) in this way, and distributing the conjugation using (13.2) and (13.3), we get:

h (13.4) (u)(γ) ∗ ρα,β(s, t, x) ∗ (−u)(γ) ∗ u(γ) ∗ s(α) ∗ (−u)(γ), u(γ) ∗ t(β) ∗ (−u)(γ) · (u(γ) · x)

1 ! Y α,β γ,χi α,β = ρχi (s, t, x) ∗ ρ (−u, ρχi (s, t, x), xi) ∗ i=n ρα,γ(−s, −u, y)−1 ∗ s(α), ρβ,γ(−t, −u, z)−1 ∗ t(β) · (u(γ) · x) = u(γ) · x for some collection of points xi, y and z. Note that by Lemma 13.5, the ρ functions above, with the α,β exception only of the ρχi (s, t, x), have only legs from E. Applying (13.2) and (13.3) repeatedly, picking up new ρ functions having legs only in E, we may reduce (13.4) to an expression of the form:

α,β (α) (β) ρ (s, t, x) ∗ ρ0 ∗ s , t ], where ρ0 is a path having only legs of E ∩ D(α, β). We may further rearrange, using commutator α,β relations, to put the path ρ (s, t, x) ∗ ρ0 in the desired circular ordering, noticing that the length of any leg not coming from E is equal to its corresponding length in ρα,β(s, t, x). This implies that α,β α,β (γ) ρχ (s, t, x) = ρχ (s, t, u · x) for any χ ∈ DE(α, β) and γ ∈ E. α,β Lemma 13.13. Let E be an ideal and α, β ∈ ∆ \ E be linearly independent. If ρχ (s, t, x) is E independent of x for all χ ∈ DE(α, β), then the action of PDE (α,β) on X factors through the action of a nilpotent Lie group.

Proof. Write DE(α, β) = {β1, β2, . . . , βr} in the circular ordering. Let G denote the factor of the group PDE (α,β) modulo the commutator relations (12.3) (which are independent of x by assumption). We ﬁrst claim that every ρ ∈ G can be written as

(β1) (βr) (13.5) u1 ∗ · · · ∗ ur

(βi1 ) (βik ) and that the elements ui are unique. Indeed, any ρ ∈ G can be written as ρ = v1 ∗ · · · ∗ vk . We may begin by pushing all of the terms from the β1 component to the left. We do this by looking at the ﬁrst term to appear with β1. Each time we pass it through, we may accumulate 76 β β β β some [u 1 , v j ] ∗ ρ(u 1 , v j ) which consists of terms without β1, since we have quotiented by the (β1) 0 0 commutator relations (12.3). So we have shown that in G, ρ is equal to u1 ∗ ρ , where ρ consists only of terms without β1. We now proceed inductively. We may in the same way push all β2 terms to the left. Notice β β now that each time we pass through, the “commutator” ρ(u 2 , v j ), j ≥ 3 has no β1 or β2 terms. Iterating this process yields the desired presentation of ρ. (β1) (βr) Uniqueness will follow from an argument similar to Lemma 5.33. Suppose that u1 ∗· · ·∗ur = (β1) (βr) v1 ∗ · · · ∗ vr , so that

(β1) (βr) (βr) (β1) u1 ∗ · · · ∗ ur ∗ (−vr) ∗ · · · ∗ (−v1) E stabilizes every point of X . Picking some a ∈ ker βr such that βi(a) < 0 for all i = 1, . . . , r − 1 implies that

β1(a) (β1) (βr) (βr) β1 (β1) (e u1) ∗ · · · ∗ ur ∗ (−vr) ∗ · · · ∗ (−e (a)v1)

E (βr) also stabilizes every point of X . Letting a → ∞ implies that (ur − vr) stabilizes every point E of X . If ur 6= vr, applying an element which contracts and expands βr yields an open set of t for which t(βr) stabilizes every point of XE, so ηβr descends to the trivial action on XE. But this implies that βr ∈ E, a contradiction, so we conclude that ur = vr. This allows cancellation of the innermost term, and one may inductively conclude that ui = vi for every i = 1, 2, . . . , r. Thus, every element of G has a unique presentation of the form (13.5). The map which assigns an r element ρ to such a presentation gives a an injective map from G to R . By Lemma 4.3, it will be continuous once its lift to PDE (α,β) is continuous. In each combinatorial cell Cβ, the map is given by composition of addition of the cell coordinates and functions ρα,β(·, ·) evaluated on cell coordinates, which are continuous. Therefore, the lift is continuous, so the map from G is continuous. So there is an injective continuous map from G to a ﬁnite-dimensional space, and G is a Lie group by Theorem 4.12. Fix a which contracts every βi. The fact that G is nilpotent follows from Lemma 12.15. α,β Lemma 13.14. If E is a maximal ideal and α, β ∈ ∆\E are linearly independent, then ργ (s, t, x) is independent of x for any γ ∈ DE(α, β) \ ∪ {α, β}.

Proof. Let Φ = DE(α, β) = {α = γ1, . . . , γk = β} be listed with the canonical circular ordering. Notice that if Φ = {α, β}, then ρα,β(s, t, x) is the trivial path for every x, so we get the result trivially. In particular, the flows ηα and ηβ commute. Now proceed by induction, proving something stronger. Namely, we assume that if |Φ| ≤ k, then the action of PΦ factors through a Lie group 2 action (this is clear in the base case, since two weights which commute generate an R action). 2 Choose a regular a ∈ R such that α(a), β(a) < 0, and let γmax be the weight of DE(α, β)\{α, β} which maximizes |γi(a)| over this set. Then γmax commutes with α, β and all γi ∈ Φ, since by induction, the weights {α, . . . , γi} and {γi, . . . , β} fit into a Lie group action and the dynamics acts by automorphisms of these Lie groups. In particular, if γmax failed to commute with one of the other γi, α or β, then we would obtain a Lie subgroup with expansion γmax(a) + γi(a) by Proposition 4.7, contradicting the maximality. This implies that every leg of the geometric commutator commutes with ηγmax , in particular that α,β γmax the functions ργi (s, t, x) are all invariant under η , and are constant by maximality of E and Lemma 13.4. Therefore, by Lemma 13.13, the action factors through a Lie group. 77 14. Homogeneity from Pairwise Cycle Structures Assuming constant commutator relations, we now construct a homogeneous structure of a Lie group for ideal factors of totally Cartan actions. Proposition 12.11, Lemma 13.13, and in the maximal factor case, Lemma 13.14, give partial homogeneous structures corresponding to certain dynamically defined subsets of weights. We carefully piece together such homogeneous structures to build one on an ideal factor XE. In particular, we use specific, computable relations between the flows of negatively proportional weights provided by the classification in Proposition 12.11. We will see that such relations yield a canonical presentation of paths, but only for a dense set of ˆ paths containing an open neighborhood of the identity in P{α,−α}. This procedure replaces a more complicated K-theoretic argument that has appeared in works on local rigidity [13, 78, 76, 77]. Our approach is to find canonical presentations for words in Pˆ using only commutator relations and symplectic relations which we assume are constant and well-defined. By fixing a regular element, we will be able to use such relations to rearrange the terms in an open set of words to write them using only stable legs, then only unstable legs, then the action (Proposition 14.12). This will imply that the quotient group of Pˆ by the commutator relations and symplectic relations is locally compact, which allows for the application of classical Lie criteria. The core of the approach is Lemma 14.11, which gives the ability to commute stable and unstable paths. These arguments allow us to give a fairly simple argument for homogeneity if the action satisfies the stronger transitivity assumption that every codimension two subgroup has a dense orbit (Sec- tion 14.1). However, this assumption is not guaranteed by the absence of rank one factors, and furthermore requires rank at least 3. The general case is much more complicated, and requires the use of the Gleason-Yamabe Lie criterion using a no small subgroups condition in Section 14.3. In the end we will have shown that if an ideal factor has constant ρ-functions, it is conjugate to a homogeneous action. In particular, this holds for any maximal factor, which sets up the induction on factoring out by a chain of ideals in Section 15.

E k Definition 14.1. Fix an ideal factor X of a topological Cartan action R y X. We say that the action has constant pairwise cycle structure if α,β (1) for each pair of nonproportional α, β, γ ∈ ∆ \ E and fixed s, t ∈ R, ργ (s, t, x) (as defined in Lemma 12.13) is independent of x and (2) for each α ∈ ∆\E such that −cα ∈ ∆ for some c > 0, the action of P{α,−cα} factors through a Lie group action.

The main goal of this section is to prove the following.

k E Theorem 14.2. If R y X is an ideal factor of a topological Cartan action with constant pairwise cycle structure, then the action is topologically conjugate to a homogeneous action.

14.1. A special case of Theorem 1.7. We ﬁrst show that we may deduce Theorem 1.7 from Theorem 14.2 under stronger transitivity assumptions on the Cartan action. It is important to note that no rank 2 action will satisfy the stronger assumptions. The general case is vastly more diﬃcult because we cannot prove constant pairwise cycle structures directly. Instead, we obtain homogeneous structures for the ideal factor actions developed in Section 13, and a priori the factor spaces are only compact metric spaces, adding extra complications. This approach gives an alternate proof of the main result of [42] with weaker assumptions (Corollary 14.5). The following proof is similar to that of Lemma 12.12.

78 k Theorem 14.3. If R y X is a topological Cartan action with k ≥ 3 such that for every pair of weights α, β ∈ ∆, the action of ker α ∩ ker β has a dense orbit, then it is topologically conjugate to a homogeneous action.

γ Proof. Notice that if a ∈ ker α ∩ ker β, then a commutes with η for any γ = sα + tβ, s, t ∈ R. In particular:

[t(α), s(β)]ρα,β(s, t, a · x) · x = a−1[t(α), s(β)]ρα,β(s, t, a · x)a · x = x. Therefore, ρα,β(s, t, a · x) = ρα,β(s, t, x) for all a ∈ ker α ∩ ker β orbits. So ρα,β(s, t, ·) is constant on the orbits of ker α ∩ ker β, and hence constant everywhere by our dense orbit assumption. Similarly, C{α,−cα} is constant on ker α orbits. Therefore, we get pairwise constant cycle structures, and by Theorem 14.2, the Cartan action is topologically conjugate to a homogeneous action. Remark 14.4. In the proof of Theorem 14.3 we used that X is a finite dimensional metric space as per definition of topological Cartan actions. When we consider ideal factor actions on the spaces XE however, we no longer know that XE is a finite dimensional metric space as the topological dimension may go up when taking the quotient. Thus we cannot use the Gleason-Palais result directly. Instead we argue below in subsection 14.3 that the group Gˆ is locally compact and has the no small subgroups property. Theorem 14.3 together with Proposition 12.6 implies the following (to see how to upgrade the topological conjugacy to a smooth one, refer to the proof of Theorem 1.7 in Section 16):

k 1,θ Corollary 14.5. If R y X is a C Cartan action on a smooth manifold X, and the action of ker α ∩ ker β has a dense orbit for every pair α, β ∈ ∆, then it is C1,θ0 conjugate to a homogeneous action. Furthermore, if the action is C∞, the conjugacy is C∞. Remark 14.6. Here we may work in C1,θ rather than C2 because we do not use the subresonant Lyapunov coeﬃcient property (see the paragraph before the statement of Theorem 12.19). In Section 15.4 we prove ideal factors have pairwise constant cycle structures through diﬀerent methods which work under weaker conditions, but require the subresonance property.

k E Standing assumption for the remainder of Section 14: R y X is an ideal factor of a topological Cartan action with constant pairwise cycle structure.

α,β 14.2. Stable-unstable-neutral presentations. We let ργ (s, t) denote the common value of α,β ργ (s, t, x) for α, β, γ ∈ ∆ \ E (which is guaranteed to be constant by constant pairwise cycle structures). Recall Deﬁnition 12.7, and let P = P∆\E (we omit the weights in E since, by construction, they act trivially on XE). Let C0 be the smallest closed normal subgroup containing all cycles (α) (β) α,β of the form [s , t ] ∗ ρ (s, t) as described in Lemma 12.13 and any element of P{α,−cα} which factors through the identity of the Lie group action provided by (2). Since such cycles are cycles at every point by assumption, C0 ⊂ C and C0 is normal. Consider the quotient group G = P/C0. k Fix a0 ∈ R , a regular element. The goal of this subsection is to show that any ρ ∈ G can be 0 (χ) reduced (via the relations in C ) to some ρ+ ∗ρ− ∗ρ0, with ρ+ having only terms t with χ(a0) > 0, (χ) ρ− having only terms t with χ(a0) < 0, and ρ0 being products of neutral elements generated by symplectic pairs (see Deﬁnition 12.10). Rather, we will show this for some subgroup obtained by combining G with the Cartan action (see Proposition 14.12). We begin by identifying well-behaved 79 subgroups of G. Given a subset Ω ⊂ ∆ \ E, let GΩ denote the subgroup of G generated by the subgroups of G corresponding to Ω. Let χ ∈ ∆ \ E be a weight such that −cχ ∈ ∆ \ E for some c, and β ∈ ∆ \ E be any linearly 0 independent weight. Let Ω = {tβ + sχ : t ≥ 0, s ∈ R}∩(∆\E), and Ω = {tβ + sχ : t > 0, s ∈ R}∩ (∆ \ E) = Ω \{χ, −cχ}.

Proposition 14.7. If Ω is as above and ρ ∈ GΩ is any element, then ρ = ρχ ∗ ρΩ0 , where ρχ ∈ G{χ,−cχ}, ρΩ0 ∈ GΩ0 . Furthermore, such a decomposition is unique. Proof. The proof technique is the same as that of Lemma 13.13. Using constancy of commutator (χ) (−cχ) relations, we may push any elements u or u to the left, accumulating elements of GΩ0 as the commutator, as well as cycles on the right (on the right because they are cycles at every 0 0 point, so we may conjugate them by whatever appears to their right). If ρχ ∗ ρΩ0 = ρχ ∗ ρΩ0 , then 0 −1 0 −1 (ρχ) ∗ ρχ = ρΩ0 ∗ ρΩ0 . But GΩ0 is a a subgroup of GΩ and it is clear that GΩ0 ∩ G{χ,−cχ} = {e}. 0 0 Therefore, ρχ = ρχ and ρΩ0 = ρΩ0 , and the decomposition is unique.

Corollary 14.8. If Ω is as above, GΩ is a Lie group. Furthermore, GΩ has the semidirect product 2 structure G{χ,−cχ} n GΩ0 , with G{χ,−cχ} isomorphic to R , the Heisenberg group, or (some cover of) PSL(2, R), and GΩ0 a nilpotent group. Proof. Notice that in the proof of Proposition 14.7, we get a unique expression by moving the elements of G{χ,−cχ} to the left, and doing so changes only the GΩ0 element. Therefore, the decomposition gives GΩ the structure of a semidirect product of G{χ,−cχ} and GΩ0 . The action of G{χ,−cχ} on GΩ0 is continuous since the action of its generating subgroups corresponding to χ and −cχ are given by commutators, which are continuous. Therefore, GΩ is the semidirect product of the Lie group G{χ,−χ} with the Lie group GΩ0 , with a continuous representation, and is hence a Lie group. The possibilities for G{χ,−cχ} follow from Proposition 12.11. The crucial tool in showing that P/C0 is Lie is to show that it is locally Euclidean. To that end, k the crucial result is Lemma 14.11. Fix a regular element a0 ∈ R , then deﬁne

+ − ∆ (a) = {χ ∈ ∆ \ E : χ(a0) > 0} , ∆ (a0) = {χ ∈ ∆ \ E : χ(a0) < 0} ,

+ − and set G+ = G∆ (a0) and G− = G∆ (a0). Consider a G{χ,−cχ} locally isomorphic to Heis or SL(2, R). Let Dχ be the subgroup of G{χ,−cχ} generated by the neutral element corresponding to the generators χ −χ η and η (see Deﬁnition 12.10). Let D ⊂ G be the product of all such Dχ. Note that the action k of each element of D coincides with the action of some element of the R action by Proposition 12.11. However, the action of D may not be faithful, as is the case for the Weyl chamber ﬂow on SL(3, R) where there are 3 symplectic pairs of weights, each generating one-parameter subgroups ∼ 2 of Diag = R . k Lemma 14.9. Suppose the R orbit of x0 is dense. Then if d ∈ D is a cycle at x0, then d is a cycle everywhere. Proof. The action of the generators of D coincides with the Cartan action, therefore the action of k D commutes with the Cartan action. Then if d is a cycle at x0, d is a cycle at any point in R (x0), k hence everywhere as the R orbit of x0 is dense. k Denote by CD the group of cycles in D at a point x0 with a dense R -orbit, and set G = G/CD and D = D/CD. Notice that G and D are topological groups since CD is a closed normal subgroup by Lemma 14.9. Furthermore, let G± denote the projections of the groups G± to G. The following 80 lemma is immediate from the fact that the action of D coincides with that of the Cartan action and Corollary 14.8:

± 0 Lemma 14.10. Let ρ ∈ G± and ρ ∈ D. Then 0 ± 0 −1 ρ ρ (ρ ) ∈ G±.

+ − Lemma 14.11. For an open set of elements ρ ∈ G+, ρ ∈ G− containing {e} × {e} there exist + 0 − 0 0 (ρ ) ∈ G+, (ρ ) ∈ G− and ρ ∈ D such that ρ+ ∗ ρ− = (ρ−)0 ∗ (ρ+)0 ∗ ρ0. Furthermore, (ρ+)0, (ρ−)0 and ρ0 depend continuously on ρ+ and ρ−.

+ − + Proof. Order the weights of ∆ (a0) and ∆ (a0) using a ﬁxed circular ordering as ∆ (a0) = − + {α1, . . . , αn} and ∆ (a0) = {β1, . . . , βm}. Since ∆ (a0) is a stable subset, G+ is a nilpotent (αn) (α1) group. Therefore, we may write ρ+ = tn ∗ · · · ∗ t1 for some t1, . . . , tn ∈ R. We will inductively + − (αn) (αk) − 0 (αk−1) (α1) 0 show that we may write the product ρ ∗ ρ as tn ∗ · · · ∗ tk ∗ (ρ ) ∗ sk−1 ∗ · · · ∗ s1 ∗ ρ − 0 0 for some ti, si ∈ R, (ρ ) ∈ G− and ρ ∈ D (all of which depend on k). Our given expression is the base case k = 1. Suppose we have this for k. If −ckαk ∈ ∆, then it must be in ∆−. Let l(k) denote the index for which βl(k) = −ckαk if −ckαk is a weight. Otherwise, since there is no weight negatively proportional to αk, we set βl(k) = −αk with l(k) a half integer so that −αk appears between βl(k)−1/2 and βl(k)+1/2. Then decompose ∆ into six (possibly empty) subsets: {αk}, {−ckαk}, ∆1 = {αl : l < k}, ∆2 = {αl : l > k}, ∆3 = {βl : l < l(k)} and ∆4 = {βl : l > l(k)}. See Figure4.

{χ : χ(a0) = 0}

∆2 αk

∆3

∆1

−ckαk ∆4

k ∗ Figure 4. Decomposing (R ) into quadrants

We let G∆i denote the subgroup of G generated by the ∆i. Notice that ∆− = ∆3 ∪{−ckαk}∪∆4 (with {−ckαk} omitted if there is no weight of this form) is stable, so again, since G− is nilpotent, − 0 (−ckαk) (ρ ) may be expressed uniquely as q3 ∗ u ∗ q4 with q3 ∈ G∆3 and q4 ∈ G∆4 (if −ckαk is not a weight, we omit this term). Now, {αk}∪∆2 ∪∆3 is a stable set whose associated group is nilpotent. (αk) 0 (αk) 0 So tk ∗ q3 = q2 ∗ (q3) ∗ s for some q2 ∈ G∆2 , (q3) ∈ G∆3 and s ∈ R. Notice that by iterating 81 k some a ∈ R for which αk(a) = 0, and β(a) < 0 for all β ∈ ∆2 ∪ ∆3, we actually know that s = tk. Thus, we have put our expression in the form:

(αn) (αk) − 0 (αk−1) (α1) 0 tn ∗ · · · ∗ tk ∗ (ρ ) ∗ sk−1 ∗ · · · ∗ s1 ∗ ρ

(αn) (αk) (−ckαk) (αk−1) (α1) 0 = tn ∗ · · · ∗ tk ∗ (q3 ∗ u ∗ q4) ∗ sk−1 ∗ · · · ∗ s1 ∗ ρ

(αn) (αk+1) 0 (αk) (−ckαk) (αk−1) (α1) 0 = tn ∗ · · · ∗ tk+1 ∗ q2 ∗ (q3) ∗ tk ∗ u ∗ q4 ∗ sk−1 ∗ · · · ∗ s1 ∗ ρ (−c α ) Now, there are three cases: there is no weight of the form −ckαk in which case u k k does not (αk) exist. Or this weight exists but commutes with tk and we simply switch the order. Otherwise, αk and −ckαk generate a cover of PSL(2, R) or Heis. In the case of PSL(2, R), if tk and u are (αk) (−ckαk) suﬃciently small, which is an open property, we may use Lemma 13.7 to commute tk and u up to an element of D. In the case of Heis, we use the standard commutator relations in Heis to commute the elements up to an element of D. Furthermore, notice that q2 and the terms appearing before q2 all belong to ∆2, so we may combine them to reduce the expression to:

0 (αn) 0 (αk+1) 0 0 (−ckαk) 0 (αk) (αk−1) (α1) 0 (tn) ∗ · · · ∗ (tk+1) ∗ (q3) ∗ (u ) ∗ (sk) ∗ d ∗ q4 ∗ sk−1 ∗ · · · ∗ s1 ∗ ρ 0 0 0 for some collection of ti, u , sk ∈ R, and d ∈ D. But by Lemma 14.10, d may be pushed to the right preserving the form of the expression and being absorbed into ρ0. We abusively do not change these terms and drop d from the expression. 0 (αk) Now, we do the final commutation by commuting (sk) and q4. Notice that {αk}∪∆1 ∪∆4 is a 0 (αk) 0 00 (αk) 0 stable subset. Therefore, we may write (sk) ∗q4 as (q4) ∗(sk) ∗q1 with q1 ∈ G∆1 , (q4) ∈ G∆4 00 and sk ∈ R. Inserting this into the previous expression, we see that the q1 term can be absorbed (αi) into the remaining product of the si terms. This yields the desired form. k Recall that the action of D coincides with the Cartan action, let f : D → R be the homomor- k phism which associates an element of D with the corresponding element of the R action. Then let Gˆ be the quotient of Pˆ (see Definition 4.6) by the group generated by ker(P → G) and elements of the form f(d)d−1. Recall that P is the free product of copies of R, and has a canonical CW-complex structure as described in Section4. The cell structure can be seen by considering subcomplexes corresponding n o (χ1) (χn) ∼ n to a sequence of weights χ¯ = (χ1, . . . , χn) and letting Cχ¯ = t1 ∗ · · · ∗ tn : ti ∈ R = R . Then a neighborhood of the identity is a union of neighborhoods in each cell Cχ¯ containing 0. Proposition 14.12. There exists an open neighborhood U of e ∈ Pˆ and a continuous map Φ: U → k ˆ G+ × G− × R such that if Φ(u) = Φ(v), then u and v represent the same element of G. + Proof. We describe the map Φ, whose domain will become clear from the definition. Let ∆ (a0) = − {α1, . . . , αn} and ∆ (a0) = {β1, . . . , βm} be the weights as described in the proof of Lemma 14.11. (χ1) (χn) Given a word ρ = t1 ∗ · · · ∗ tn , we begin by taking all occurrences of αn in ρ and pushing them to the left, starting with the leftmost term. When we commute it past another αi, we accumulate α ,αn only other αj, i + 1 ≤ j ≤ n − 1, in ρ i , which we may canonically present in increasing order on the right of the commutation. A similar statement holds for the commutation of αn with βi. We iterate this procedure as in the proof of Lemma 14.11 to obtain the desired presentation. Since the commutation operations involved are determined by the combinatorial type, the resulting (χi) presentation is continuous from the cell Cχ¯. Furthermore, if one of the terms ti happens to be 0, 82 the procedure yields the same result whether it is considered there or not. Thus, it is a well-defined k ˆ continuous map from P → G+ × G− × R (it is continuous from P because it is continuous from each Cχ¯). Notice that in the application of Lemma 14.11, we require that st 6= 1 whenever we try to pass t(χ) by s(−χ). This is possible if s and t are sufficiently small. Thus, in each combinatorial pattern, since the algorithm is guaranteed to have a finite number of steps and swaps appearing, and each (χi) term appearing will depend continuously on the initial values of ti , we know that for each χ¯, some neighborhood of 0 will be in the domain of Φ, by the neighborhood structure described above. k Notice that the reduction of a word u to a word of the form u+ ∗ u− ∗ a ∈ G+ × G− × R uses only relations in Gˆ. Therefore, if after the reductions, the same form is obtained, the original words must represent the same element of Gˆ. Remark 14.13. Unfortunately, it is not at all clear from Proposition 14.12 that the map Φ descends to a map from a neighborhood of e ∈ Gˆ, in which case we could directly apply Theorem 4.12. While at first glance it appears to give local coordinates on Gˆ, it is only half of the story. That Φ(u) = Φ(v) implies that u and v represent the same element of Gˆ only implies that if the map were to descend, then it would be injective. Therefore, we must go through another argument to show that Gˆ (or rather, yet another factor, which we will call G˜) is a Lie group. This is done in Section 14.3. 14.3. No Small Subgroups Argument. In this subsection, we prove that XE is the homogeneous space of a Lie group in Proposition 14.14. First we note that Gˆ is locally compact in Lemma 14.15. Then we show that small compact subgroups of Gˆ act trivially on XE employing the hyperbolic k dynamics of the R action. This allows us to apply standard results from Hilbert’s 5th problem to the image G˜ of Gˆ in the homeomorphisms of XE. Proposition 14.14. The action of Gˆ on XE factors through a connected Lie group G˜. In particular, E k k X is the homogeneous space of G˜, and R acts via the image of R in G˜. The rest of this section is devoted to a proof of Proposition 14.14. Lemma 14.15. Gˆ is locally compact. k Proof. Choose a compact neighborhood C of (e, e, e) ∈ G+ × G− × R . Then using a circular ordering on the positive and negative weights, one sees that C maps into Pˆ in the obvious way, f : C → Pˆ. Then Φ(f(C)) = C (where Φ is as in Proposition 14.12). But then the projection of C onto Gˆ contains a neighborhood, since any element sufficiently close to e ∈ Pˆ can be reduced to an element in f(C) using relations from Gˆ. Since C is compact, we have constructed a compact ˆ neighborhood of e ∈ G. Since Gˆ is connected, locally compact and acts transitively on XE, Proposition 14.14 immediately follows from the following result: Proposition 14.16. Let N ⊂ Gˆ be the kernel of the action of Gˆ on XE, ie, n o N = g ∈ Gˆ : g · x = x for every x ∈ XE .

Then Gˆ0 := G/Nˆ is locally compact, has the no small subgroups property, and hence is a Lie group. We will prove this proposition by the chain of arguments below. We will need to proceed very carefully with the order in which we choose various gadgets. Since N is closed, Gˆ0 is locally compact and Hausdorﬀ. 83 First recall that Gˆ+ and Gˆ− are Lie groups by 13.13, and hence do not have small subgroups. ˆ0 ˆ0 E k We denote by G+ and G− their images in the homeomorphism group of X , and the image of R by A. Then they are nilpotent Lie groups as quotients of Lie groups by closed normal subgroups ˆ0 ˆ0 0 ˆ0 are Lie groups again. Also note that G+, G− and A generate G . ˆ ˆ0 We ﬁrst describe the kernel K+ of the projection G+ → G+.

Lemma 14.17. K+ = {e}. + E Proof. Suppose g ∈ Gˆ+ acts trivially on X , and recall that Gˆ+ is a nilpotent Lie group whose + Lie algebra is given as a sum of 1-dimensional Lie algebras for each χ ∈ ∆ (a0) \ E ⊂ ∆ \ E. + P λ k −1 + −1 Write g = exp v with v = λ/∈E cλv . For any b ∈ R , b g b ∈ K+ as well. Pick b so that conjugation by b contracts Gˆ+, and so that all λ(b) are distinct and not 0. Note that the cyclic + n λ group generated by g conjugated by b will converge to the image of the one-parameter group ηs λ E where λ is the weight with λ(b) the smallest λ(b) with cλ 6= 0. Hence ηs ⊂ ker+. But this implies λ E that ηs fixes each M (x) ⊂ X, so λ ∈ E. This contradicts to our assumption that λ 6∈ E. k E ˆ0 Lemma 14.18. There is an R -periodic point p ∈ X such that G+ has a free orbit on p. E k + + ˆ0 Proof. Suppose first that p ∈ X is periodic for the R action, and that g p = p for some g ∈ G+. + P λ k By Lemma 14.17, g = exp v with v = λ/∈E cλv . Pick b ∈ R in the stabilizer of p which ˆ0 shrinks G+ and for which all the weights λ(b) are all distinct. Then the cyclic group generated by + n λ ˆ ˆ0 g conjugated by b will converge to the image of a one-parameter group ηs ⊂ G+ in G+E, with ˆ0 λ∈ / E. Hence we get a one parameter subgroup of G+ which fixes p. Now suppose that every periodic point p is fixed by a one-parameter subgroup ηβ for some β∈ / E. Let F β denote the fixed point set of the whole one-parameter group ηβ. Since the periodic points β E are dense (by Theorem 5.13), the union ∪βF = X of these finitely many closed sets is all of E k X . Hence at least one of them has non-empty interior. Since the R action has dense orbits and normalizes each ηβ, F β = XE for some weight β. This implies that β ∈ E, in contradiction to our k ˆ0 choice of β. Therefore there is an R periodic point p on which G+ acts freely. E Let dE be the Hausdorff metric on X (descended from the metric in the definition of topological ˆ0 Cartan action, see Lemma 13.11), and d+ be a right-invariant metric on on G+. Let p be as in Lemma 14.18. Lemma 14.19. For all δ > 0 and C > 1 there is a constant > 0, and a neighborhood V of p ˆ0 such that for g in the compact annulus Aδ,C in G+ defined by δ ≤ d+(g, 1) ≤ Cδ and all x ∈ V , ˆ0 dE(x, gx) > . Moreover, G+ acts locally freely on V . ˆ0 E Proof. Let f(g, x) = dE(g · x, x). Notice that f is a continuous map from G+ × X → R, and ˆ0 f(g, p) > 0 for all g ∈ G+ \{e}. Therefore, for any δ, f|Aδ,C ×{p} ≥ 1 for some 1 > 0. Since f is continuous, there exists a neighborhood V such that f(g, x) > = 1/2 for all x ∈ V and g ∈ Aδ,C . ˆ0 ˆ0 Local freeness of the action of G+ in a neighborhood of p now follows easily. Since G+ is a Lie l group, if h is very close to e (depending on how close C is to 1), then h ∈ Aδ for some l ∈ N. l Hence if hv = v for some v ∈ V , also h v = v in contradiction to the freeness of Aδ,C acting on V . ˆ0 It follows that G+ acts locally freely on V , as desired. Our last technical ingredient in the proof of Proposition 14.16 is the following: Lemma 14.20. There exists a neighborhood U ⊂ Gˆ of e such that if K is a compact normal subgroup contained in U, then K acts trivially on XE. 84 k ˆ0 Proof. Pick a periodic point p, and let c ∈ R fix p and contract G+ · p. That is, we suppose that ˆ0 −1 −1 conjugation by c on G+ is a contraction, so d(cgc , e) < C d(g, e) for some C > 1. Fix a small δ > 0, and let > 0 and V be a neighborhood of p be as in Lemma 14.19. We will also assume that U is the image under Φ (cf. Proposition 14.12) of three balls (w.r.t. the exponential maps) in ˆ ˆ k 3 G+, G− and R , and that d(up, p) < /100 for all u ∈ U . Now suppose K is a compact normal subgroup contained in U. Let k ∈ K and write k = k+k−a + ˆ − ˆ k + with k ∈ G+, k ∈ G− and a ∈ R (as guaranteed by Proposition 14.12). If k 6= 1, pick l ∈ N l + −l such that c k c ∈ Aδ,C . Then clk+c−l = (clkc−l)a−1(cl(k−)−1c−l). l + −l The left hand side moves p by at least since c k c ∈ Aδ. For the expression on the right hand side, first note that clkc−l ∈ K ⊂ U, a−1 ∈ U since U is symmetric, and finally cl(k−)−1c−l ∈ U since conjugation by c contracts balls in U −. By the assumption on the neighborhood U of 1 in Gˆ, d((clkc−1)a−1(cl(k−)−1c−1) · p, p) < /100. This is a contradiction, and therefore, for all k ∈ K, + − k k k = 1. By a similar argument, also k = 1, and hence K ⊂ R . Since R does not have small subgroups, K = {e}. Proof of Proposition 14.16. Let U be as in Lemma 14.20. By the Gleason-Yamabe theorem (Theo- rem 4.11), there exists a compact normal subgroup K ⊂ U such that G˜ = G/Kˆ is a Lie group. But ˆ0 K ⊂ N by Lemma 14.20. Therefore, G is a factor of a Lie group, and hence a Lie group. By Proposition 14.14, XE is the homogeneous space of a Lie group and hence a manifold. Theorem 14.2 is now immediate from the following:

Proposition 14.21. Let Y be a compact metric space, G be a Lie group, and suppose that G y Y is a transitive action of G. If there is a subgroup A ⊂ G such that the restriction of the action of G to A is an ideal factor of a topological Cartan action with pairwise constant cycle structure, ◦ then StabG(x) , the identity component of StabG(x), is normal in G. Furthermore, the action of ◦ H = G/ StabG(x) is locally free, Γ = StabH (x) is cocompact in H, and Y carries a unique smooth structure for which the action of H is smooth and the map H/Γ → Y given by gΓ 7→ g · x is a C∞ diffeomorphism. Proof. It is immediately clear from standard Lie theory that Y can be endowed with a unique smooth structure making it diffeomorphic to G/ StabG(x). First, notice that if n k o A0 = a ∈ R : a · y = y for every y ∈ Y ⊂ StabG(x), then A0 is a closed, normal subgroup of G. Therefore, H = G/A0 is a Lie group. We claim that the action of H on Y is locally free. It suffices to show local freeness at one point, since the action is transitive and the orbits are intertwined. We consider the action of A1 = A/A0. Let x be a point with a dense A1-orbit (which exists since the action is the factor of a transitive ◦ action), and assume that Bx = StabA1 (x) ⊂ A1 is nontrivial. Then since Bx commutes with A1, Bx stabilizes every point of A1 · x, and therefore all of Y by continuity and density of A1 · x. Therefore, Bx is trivial. Since the local freeness of A1 is an open property, and it holds at every dense A1 orbit, it must 0 0 hold on an open dense set. By Lemma 14.19, the set of points for which G+ and G− act locally k freely is open, nonempty and R -invariant, so again it is open and dense. Therefore, we may find a 0 0 periodic orbit for which the actions of A1, G+ and G− are all locally free, call it p. We claim that the action of H at x is locally free. Indeed, if not, there exists a one-parameter subgroup stabilizing x. Let v denote its generator. Notice that, by locally transverse laminations, 85 P χ χ Lie(G) is written as the (possibly not direct) sum of subalgebras Lie(A) + χ∈∆ E , where E χ is the algebra of the one parameter subgroup ηt . Then v ∈ Lie(Γ), and we may write v as v = u + P vχ ∈ Lie(G), with u ∈ Lie(A) and vχ ∈ Eχ. Then if a · p = p, a exp(tv)a−1 ∈ Γ. Therefore, Ad(a)nk v converges to an element of Lie(Γ). Now, notice that if v has any unstable components, ||Ad(a)nk v|| Ad(a)nk v lim ∈ Lie(G+), k→∞ ||Ad(a)nk v|| and that by choice of x, the action of G+ is locally free. This implies that the unstable component of v is trivial. Similarly, the stable component of v is trivial. Finally, x was chosen so that the action of A1 is locally free at x.

15. Extensions by Maximal Ideals k We work in the following setting: Assume that R y X is a topological Cartan action satisfying E the assumptions of Theorem 12.19, E0 ⊂ ∆ is some ideal for which the induced action on X 0 is homogeneous, and E ⊂ E0 ⊂ ∆ is the largest ideal strictly contained in E0. Set ∆b = ∆ \ E0 and E0 ∆f = E0 \E. We call ∆b the base weights and ∆f the fiber weights. Then X is a quotient space of E E0 E E X , and ∆f are all flows along the “fibers” (i.e. the sets M (x)/M (x)), and ∆b are flows on X E0 covering the homogeneous flows on X . Furthermore, [∆, ∆f ] ⊂ ∆f , and we have the following important feature coming from maximality (compare with Lemma 13.4):

E χ Lemma 15.1. Suppose that h : X → R is a continuous function invariant under η for some E E0 χ ∈ ∆f . Then h is invariant along the ﬁbers of X → X .

E Proof. Notice that since h is deﬁned on X , then f lifts to a function h˜ : X → R which is invariant under every α ∈ E. By assumption, h˜ is also invariant under ηχ, so f is invariant under the weights of E ∪ {χ}. Then, by deﬁnition, h˜ is invariant under the ideal closure E ∪ {χ}. By maximality of E, E ∪ {χ} = E0. Therefore, h˜ is invariant along the weights of E0, and so is h. This completes the proof. The main result of this section is the following.

k Theorem 15.2. Let R y X, E0 and E be as discussed above. Then the induced action on the ideal factor XE has constant pairwise cycle structure.

Part (2) of Deﬁnition 14.1 follows from the higher rank assumptions and Lemma 12.12. Therefore, α,β we must show (1). That is, we must show that if α, β ∈ ∆ are linearly independent, then ρχ (s, t, x) is independent of x for χ 6∈ E. We do this in three parts: if α, β ∈ ∆f (Section 15.2), if α ∈ ∆b and β ∈ ∆f (Section 15.3), and if α, β ∈ ∆b (Section 15.4).

E0 Remark 15.3. While we know that if α, β ∈ ∆b, they ﬁt into the action of a Lie group on X , it is not clear that this Lie group action lifts to XE. This means that the “base-base” relations of Section 15.4 are not immediate from the homogeneous structure. This is realized in examples, for k instance in Anosov Z action on nilmanifolds which are not tori. In this case, the maximal factor is N a torus, and the corresponding Lie group whose action gives a homogeneous space structure is R . N k But the R action will not lift to the nilmanifold. One can obtain an R action example by taking a suspension.

We begin by showing that the action of the weights in ∆f are transitive on each fiber. 86 k 15.1. Structure of fibers. Consider a topological Cartan action of R on a space X with locally transverse laminations as in Definition 12.2. We will show that the fibers of the natural projection F maps πF : X → X are endowed with totally Cartan actions themselves, and their coarse Lyapunov foliations come from those on the total space, inheriting their properties and metrics. This allows us to use the arguments of Section 14 on the fibers in Section 15.2. Given an ideal F in the set of weights, we form the subgroup PF ⊂ P generated by the legs of the weights in F (recall Definition 12.7). Proposition 15.4. Let F be an ideal, and XF the associated ideal factor, with natural projection F F λ F F map π = π : X → X . Suppose that the flows ηs for λ ∈ ∆ \ F on X generate a Lie group H . −1 Then the preimages π (x) are the orbits of PF . k k Moreover, let p be a periodic point of the R -action on X. Then the stabilizer of p in R is l k−l −1 isomorphic to Z × R , for some 0 ≤ l ≤ k. Its action on the fiber π (p) is a topological Cartan action with locally transverse laminations. We note that transitive topological Cartan actions always have a dense set of periodic points by Theorem 5.13.

Proof. Order both the weights in F by some arbitrary order η¯ = {η1, . . . , ηn1 } and ∆ \ F by ¯ ¯ ¯ δ = {δ1, . . . , δn2 }. This induces an order β = (¯η, δ) of all weights by putting the weights in F first. k Since the action has locally transverse laminations, the map from Ψβ¯ : Cβ¯ ×R → X is locally onto. k k F Let V ⊂ R be the identity component of the elements v ∈ R such that v · x ∈ M (x). Then V k F is a vector subspace of R and if w 6∈ V , then for sufficiently small t ∈ R, tw · x 6∈ M (x). Let W be any subgroup transverse to V . Note that Cβ¯ = Cη¯ × Cδ¯ and that π is constant on sets of the form ((V × {0}) × (C × {ρ})) · x. The assumption that HF is a Lie group implies that Ψ | η¯ β¯ W ×Cδ¯ is a continuous local injection (this follows from transversality of subgroups in Lie groups). Hence CV ×η¯ maps onto a neighborhood in the fiber, since nearby fibers are locally connected uniquely by a path in δ¯, and η¯ moves within fibers. Hence the fibers are locally path connected. We claim the fibers are path connected, which will follow from their description in Proposition 13.6. It is easy to see that at each step of the induction, the set Si(x) is path connected. Therefore, the limit of the induction, the fiber M F (x), is also connected. Since each fiber is locally path connected, compact and connected, any pair of points can be joined using a chain of points x1, . . . , xn F such that xi+1 ∈ Cη¯ · xi. In particular, the fibers are exactly the orbits of P and have locally transverse laminations. F k F l k−l Next consider a periodic point p ∈ X . The stabilizer of p in R y X is a subgroup Z × R which acts on the fiber π−1(p). The latter is a compact metric space of finite dimensions as it is a closed subspace of the finite dimensional space X. The remaining conditions of Definition 12.2 are easy to check. F F Corollary 15.5. Let F ⊂ F0 ⊂ ∆ be two ideals, and X and X 0 the associated ideal factors, with natural projection map π = πF : XF → XF0 . Suppose that the flows ηλ for λ ∈ ∆ \ F on XE0 F0 s 0 generate a Lie group HF0 . Then the preimages (πF )−1(x) are the orbits of PF0 acting on XF . F0 k F k Moreover, let p be a periodic point of the R -action on X 0 . Then the stabilizer of p in R is isomorphic to l × k0 , for some k0. Its action on the fiber (πF )−1(p) is an ideal factor of a Z R F0 topological Cartan action with locally transverse laminations. 15.2. Fiber-Fiber Relations. First we prove constancy of fiber-fiber relations. Given two weights α and β, constancy of the function ρα,β(s, t, ·) along each fiber can be established in a manner similar to the maximal factor, using Corollary 15.5, Lemma 13.14 and Theorem 14.2. This will give 87 a homogeneous structure to each fiber, and by using the intertwining property of the action and topological transitivity of the factor action on the base, we will conclude that these functions are constant everywhere. We also will obtain a classification of the groups which may appear, which α,β will be important in establishing constancy of ρ (s, t, ·) when α and/or β are in ∆b.

Proposition 15.6. ∆f generates the action of a ﬁxed Lie group H which acts transitively on each E E set M 0 (x) ⊂ X . Furthermore, H is either abelian, 2-step nilpotent or R-split semisimple, and α for every α ∈ ∆f every η -invariant function is also H-invariant.

E0 α,β Proof. To get a Lie group action on each M (x), observe that if α, β ∈ ∆f , ρ (s, t, x) is constant along M E0 (x) by Lemma 13.14 and Lemma 15.1. We also know from Lemma 12.12 that the symplectic relations factor through a Lie group action. Hence, we obtain a Lie group action on each fiber (possibly depending on x) using the arguments of Section 14. Let Hx denote the Lie group which acts on M E0 (x). k We now claim that if x0 has a dense R orbit, Hx0 acts on every fiber (and in particular, Hx E is a factor of Hx0 for every x). Notice that if α ∈ ∆f , there exists at least one x ∈ X and α some ε > 0 such that ηt (x) 6= x for t ∈ (−2ε, 2ε), since the fibers are homogeneous spaces of Lie α groups Hx. Therefore, by continuity, in some neighborhood of x, ηt (x) 6= x for t ∈ (−ε, ε). Since k α E the R orbit of x0 enters this neighborhood, we get that ηt does not fix M (x0), and therefore generates a one-parameter subgroup of Hx0 . Let Xα(x) denote the element of Lie(Hx) generating α k η when x ∈ R · x0. Then [Xα(x),Xβ(x)] = cαβ(x)Xα+β(x) and [Xα(x),X−cα(x)] = 0 if c 6= 1 and k [Xα(x),X−α(x)] = Yα(x) for some Yα(x) ∈ R . Pushing these algebra relations forward, it is clear k α(a)+β(a) that cαβ(x) is constant along R orbits (since both sides rescale by e ) and Yα(x) is constant k along R -orbits for similar reasons. Therefore, there is an isomorphism of the Lie groups Hx and k α Hy when x and y belong to the same R -orbit of x0 which intertwine the actions of the η , α ∈ ∆f . k So Hx0 acts on every fiber in the R -orbit of x0, and extends to the entire space continuously. Let

H denote the group Hx0 , which acts on each fiber. This gives a fixed Lie group H acting transitively on each fiber M E0 (x) ⊂ XE. Notice that H β is generated by the one parameter subgroups Uβ corresponding to the coarse Lyapunov flows η , k−l L β ∈ ∆f . If gβ = Lie(Uβ), one sees from Lemma 14.12 that Lie(H) = R ⊕ β∈∆ gβ, since the stable, unstable and acting groups give a local product structure to H. Let x ∈ XE0 be a periodic ∼ k−l l orbit of the homogeneous Cartan action, and V = R × Z be the stabilizer of the point. Then V β E acts on the fiber of x by a Cartan action, and for every β ∈ ∆f , any η -invariant function from X is constant along the fibers M E0 (x) by Lemma 15.1. Fix a periodic point p for the action of V on the fiber containing x0. Let the homogeneous space above x be given by H/Γx, with Γx ⊂ H a closed subgroup. We can then write the action of a ∈ V k l as a · (gΓx) = (ϕ(a)ψa(g))Γ, where ϕ : R × Z → H and ψa is an automorphism of H which fixes Γx. Note that Γx is discrete in H by Proposition 14.21. So we may apply Theorem 6.1. If H has a semisimple factor, then the closure of the coarse Lyapunov subgroups in such a factor are contained in some section, since the lattice splits as a semidirect product as well. Therefore, since each coarse Lyapunov subgroup must have a dense orbit in the fiber by maximality of the ideal chosen, H is semisimple and StabH (x) must be an irreducible lattice for every x ∈ XE. If H does not have a semisimple factor, then H is a solvable group. But each coarse Lyapunov flow is contained in the nilradical by Theorem 6.1(4). If H were not nilpotent, it would have an abelian factor, which is part of the acting group. By definition of the sets M E(x) (Definition 13.1), and the fact that lattices intersect the nilradical in a lattice (Theorem 6.1(4)), one sees that H is nilpotent (if the group is solvable, points projecting to different elements of the abelian factor can 88 be separated by functions invariant under the nilradical). Furthermore, exp([h, h]) has closed orbits. 0 0 Let E denote the set of weights in [h, h], so E ∪ E ⊂ E0 has is an ideal. By maximality of the ideal 0 k chosen, E = ∅. In particular, exp([h, h]) must be contained in the orbit of R , and is generated by commutators of symplectic pairs of weights. Since h is nilpotent, we know the group generated by any given pair of symplectic weights must be either abelian or the Heisenberg group by Proposition 12.11. We know that [[h, h], h] ⊂ [h, h]. For each coarse Lyapunov subalgebra gβ, [[h, h], gβ] ⊂ gβ, since we have shown that [h, h] generates part of the k action. But L gβ ∩ [h, h] = {0}, so R β∈∆f [h, h] is central in h and h is a 2-step nilpotent group. k Remark 15.7. It is easy to see that if any part of the R action is contained in the fiber, then its corresponding one-parameter subgroup cannot have dense orbit on the total space. Therefore, if one assumes that every one-parameter subgroup has a dense orbit, one may conclude that H is abelian. 15.3. Base-Fiber Relations. We will now analyze commutator cycles between weights from the base and the fiber.

α,β Proposition 15.8. If α ∈ ∆b and β ∈ ∆f are linearly independent, then ρ (s, t, x) is independent of x. Recall that by Proposition 15.6, there exists a Lie group H generated by the ﬁber subgroups acting transitively on each ﬁber, and that H is either semisimple, abelian or 2-step nilpotent. We prove Lemma 15.8 by treating these cases separately.

Case: H is semisimple. We first address the case when H is semisimple. In this case, the β flows η with β ∈ ∆f generate the root subgroups of H. Furthermore, as in Proposition 12.11, the translation action by some Cartan subgroup of H coincides with the restriction of the Cartan action k to the elements preserving the fiber H · x. Let V ⊂ R be the subspace which preserves the fibers, so that dim V = rankR(H). k Now if a ∈ R is any element, it still induces an automorphism of H. Since this automorphism k k varies continuously with the choice of a ∈ R and the automorphism at e ∈ R is the identity, the induced automorphism must be homotopic to the identity. Furthermore, it preserves the root spaces by Proposition 4.7. The space of such automorphisms can be identified with V , since it is exactly the inner automorphisms determined by these elements since H is R-split. Therefore, if W = T ker β, then W ∩ V = {0} and dim(W ) = k − rank (H). Therefore, k = V ⊕ W . Now β∈∆f R R notice that if a ∈ V and α ∈ ∆b, then α(a) = 0, since a only moves the position in the fiber and α is transverse to the fiber. Similarly, if β ∈ ∆f , β(a) = 0 for every a ∈ W , by definition of W . α,β Assume for contradiction that α ∈ ∆b, β ∈ ∆f , but ρ (s, t, x) 6≡ 0. Then there exists some χ ∈ ∆ such that χ = sα + tβ with s, t > 0. But neither χ|V nor χ|W is identically 0, so it can’t be α,β in either ∆b or ∆f . This is a contradiction, so we may conclude ρ (s, t, x) ≡ 0.

Case: H is abelian. In this case, we will prove the following stronger claims:

(15.1) if [α, β] 6= ∅, [α, β] = {β + α, . . . , β + mα} for some m ≥ 1 α,β k E (15.2) ρβ+kα(s, t, x) = cs t for all s, t ∈ R and x ∈ X

We prove these by an induction on on |D(α, β) ∩ ∆f |. Before starting, we need to develop some preparatory material.

α,β α,β E0 Lemma 15.9. If α ∈ ∆b and β ∈ ∆f , then ρ (s, t, x) = ρ (s, t, y) if y ∈ M (x). 89 α,β Proof. Notice that if χ ∈ ∆b, then ρχ ≡ 0, since β ∈ ∆f = E0 \E and E0 is an ideal. Therefore, as in Lemma 13.14 and by Lemma 15.1, it suffices to show that there exists χ0 ∈ D(α, β)∪{β} such that χ0 χ {χ0} η commutes with η for every χ ∈ (D(α, β) ∩ ∆f ) ∪ {α} ∪ {β} (such a χ0 will satisfy M (x) = E0 M (x) by maximality of E ⊂ E0). Choose χ0 to be the weight closest to α in (D(α, β)∪{β})∩∆f . Then since the fiber is abelian, it commutes with everything in (D(α, β) ∪ {β}) ∩ ∆f , and since it α,β is the closest weight to α, it commutes with α. Thus, ρ (s, t, x) is constant along the fiber.

(β) t2 s(α) (β) t1

y (α) (β) s t1 (β) t2

(β) x t1 s(α)

α,β Figure 5. The cocycle property for ρχ0 (s, ·, x)

α,β Lemma 15.10. In the case of H abelian, ρ (s, ·, x): R → H is a homomorphism (which may depend on s, x).

Proof. From the definition of geometric commutators and the fact that [χ0, α] = ∅, we get that α,β α,β α,β ρ (s, t1 + t2, x) = ρ (s, t1, x) + ρ (s, t2, y) for some y in the same fiber of x (c.f. Figure5). Notice that in the figure, the arrow points in the α direction, which is the only direction along the base. The blue curves represent the geometric commutators evaluated at their respective points. Since the fiber is abelian, and the “rectangle” of blue segments and black segments based at y is contained in the fiber, it is exactly a rectangle. Therefore, we get the desired additivity.

Lemma 15.11. If χ ∈ [α, β], then χ = uα + β for some v ∈ R+. k Proof. Recall that if [α, β] ⊂ D(α, β), so χ = uα + vβ with u, v > 0. By (12.4), if a ∈ R ,

uα(a)+vβ(a) α,β α,β α(a) β(a) e ρχ (s, t, x) = ρχ (e s, e t, a · x). Choose x to be a periodic orbit. Notice that χ ∈ ∆f and that χ 6= −cβ. Therefore, |∆f | ≥ 3, since it cannot be just one and having only two weights implies it must be a symplectic pair {χ, −cχ}. Hence, the action on the ﬁber has a rank two subgroup acting faithfully (otherwise a 90 coarse Lyapunov has dimension larger than 1). Therefore, {β(a): a · x = x, |α(a)| < ε} is dense in vr α,β α,β r R. By continuity and letting ε → 0, e ρχ (s, t, x) = ρχ (s, e t, x). By Lemma 15.10, we conclude that v = 1. Lemma 15.11 implies that [α, β] ⊂ {uα + β : u > 0} ∩ ∆. We inductively show that each α,β k ρχ (s, t, x) is a polynomial of the form cs t for some c independent of x for all χ ∈ [α, β]. Write [α, β] = {χ0, χ1, . . . , χm} = {u0α + β, u1α + β, . . . , umα + β}, with u0 < ··· < um. α,β For any χ ∈ [α, β] written as χ = uα + β, let ϕχ(s, x) = ρχ (s, 1, x). Then by (12.4), χ(a) α,β α(a) β(a) e ϕχ(s, x) = ρχ (e s, e , a · x), and

−uα(a) α(a) (15.3) ϕχ(s, x) = e ϕχ(e s, a · x), k α,β for every a ∈ R by linearity of ρχ in the second variable. Recall that by b) of the higher rank k assumptions, either ker α has a dense orbit, or there is a factor π : X → R/Z of the R action, ker α preserves every ﬁber, and has a dense orbit in every ﬁber.

Lemma 15.12. The function ϕχ(s, ·) depends only on its ker α closure. More precisely, one of the following holds:

(1) If ker α has a dense orbit, then ϕχ(s, x) is independent of x for every s. (2) If ker α has orbits dense in the ﬁbers of some circle factor, ϕχ(s, x) depends only on s and π(x).

k Proof. Notice that by (15.3), if a ∈ ker α, then ϕχ(s, x) = ϕχ(s, a · x) for every a ∈ R . There- fore, since ϕχ is continuous in s and x, it is constant on ker α closures. This implies the results immediately.

Let Φ = {α} ∪ (D(α, β) ∩ ∆f ) be the weight α together with all ﬁber weights between α and β.

Lemma 15.13. If ker α has a dense orbit, PΦ factors through a nilpotent Lie group action H y X. If ker α has orbits dense in the ﬁber of a circle factor, then for every ξ ∈ R/Z, PΦ factors through −1 a Lie group action Hξ y π (ξ).

Proof. In case (1) of Lemma 15.12, ϕχ(s, x) is independent of x for every s. Therefore, by Lemma 15.10, given s, t ∈ R:

α,β α,β ρχ (s, t, x) = tρχ (s, 1, x) = tϕχ(s, x) is independent of x. In case (2) of Lemma 15.12, notice that it depends only on π(x), and π(x) is constant along PΦ-orbits since any coarse Lyapunov leaf projects trivially onto any circle factor (since the circle factor is isometric, and distinct points on the same coarse Lyapunov leaves see exponential expansion and contraction). Therefore, by considering the action on X if ker α has a −1 dense orbit, or on π (ξ) otherwise, Lemma 13.13 implies that the action of PΦ factors through a nilpotent Lie group action (on the respective spaces). α,β u Corollary 15.14. If χ ∈ D(α, β), then χ = uα + β, with u ∈ Z+ and ρχ (s, t, x) = cχs t for some cχ independent of x.

Proof. By Lemma 15.13, the action of PΦ factors through the action of a Lie group, but the action may depend on which fiber of the circle factor one lies in. If x ∈ X, let ξ = π(x) denote its α,β u coordinate on the circle factor (if it exists). By Lemma 12.15, ρχ (s, t, x) = cξs t, and u ∈ N. This finishes the proof if there is no circle factor. If there is a circle factor, by (15.3), 91 α,β uα(a) −α(a) (15.4) ρ (s, t, a · x) = tϕχ(s, a · x) = te ϕχ(e s, x) uα(a) −α(a) u u α,β = te cπ(x)(e s) = cπ(x)ts = ρ (s, t, x). k So cπ(a·x) = cπ(x) for all a ∈ R , and hence the coefficient is constant. Case: H is 2-step nilpotent. Observe that in the abelian case, the only weights χ for which α,β ρχ 6≡ 0 are those in D(α, β). But D(α, β) contains no negatively proportional weights, so the weights of D(α, β) generate an abelian subgroup of H which does not contain the center of H (since the center is always generated by symplectic pairs α and −α). In particular, the arguments for the abelian case work verbatim in the 2-step nilpotent case. 15.4. Base-base relations. For the final step in proving constant pairwise cycle structure, consider two weights α and β from the base. While we already know that the geometric commutator is constant when taken along the base, the same is not at all clear on the space XE as the geometric commutators may and often will involve weights from the fibers. The following proposition is the main result in this section, and will finish the proof of Theorem 15.2.

α,β Proposition 15.15. If α, β ∈ ∆b are nonproportional, then ρ (s, t, x) is independent of x. α,β When α, β, χ ∈ ∆b, the functions ρχ (s, t, x) are deﬁned on the total space and were proved to α,β descend to the ideal quotient in Lemma 13.12, so it suﬃces to show that ρχ (s, t, x) is independent of x for every α, β ∈ ∆b and χ ∈ ∆f . As in Section 15.3, the proof of this proposition is split into the semisimple and abelian case.

Case: H is semisimple. We use the results and notations from the semisimple case in Section k 15.3. Recall that W ⊂ R is the intersection of all the kernels of the fiber weights, and that the homogeneous action of W is faithful on the base XE0 . Given nonproportional base weights α, β ∈ ∆b, find a ∈ W with both α(a), β(a) < 0. Then any leg λ in the geometric bracket [α, β] in XE gets contracted by an, n → ∞. Since a is in the kernel of all fiber weights, no λ ∈ [α, β] can be a fiber weight.

Case: H is abelian. The proof of this case involves several inductions. The outermost induction is on #(D(α, β) ∩ ∆b). We unify the proof of the inductive and base cases, and point out the part of the proof where #(D(α, β) ∩ ∆b) = 0 is used for the base case. In each step we will show

(15.5) if χ ∈ [α, β], then χ = kα + lβ for some k, l ∈ Z+ α,β k l (15.6) if χ = kα + lβ ∈ [α, β], then ρχ (s, t, x) = cχs t α,β First note that if χ ∈ ∆b, then ρχ (s, t, x) is independent of x, since the bracket relations on the total space XE project to bracket relations on XE0 . Since XE0 is homogeneous, Claims 15.5 and 15.6 follow immediately for χ ∈ [α, β] ∩ ∆b. Therefore, it suﬃces to show the claims for every χ ∈ [α, β] ∩ ∆f . We now begin the second induction. Let Ωl = {uα + vβ : u + v = l, u, v ≥ 0} ∩ ∆f . Then there are ﬁnitely many values l0 < l1 < ··· < lm such that m [ [α, β] ∩ ∆f ⊂ D(α, β) ∩ ∆f = Ωli . i=0 92 S 0 Given a subset S of weights, and a weight χ ∈ ∆, let [χ, S] = χ0∈S[χ, χ ].

Lemma 15.16. If χ ∈ [α, Ωl] or [β, Ωl], then χ ∈ Ωl+t for some t ≥ 1.

Proof. Without loss of generality consider χ ∈ [α, Ωl]. Notice that χ ∈ ∆f , since [α, ∆f ] ⊂ ∆f , and each Ωl ⊂ ∆f . Thus, the lemma follows from Claim 15.1 of Section 15.3. S Since [α, β]∩∆f = Ωli , it suﬃces to show Claims 15.5 and 15.6 for each weight χ ∈ Ωli . We will do this by an induction on i, starting from i = −1, setting Ωl−1 = ∅. Assume Claims 15.5 and 15.6 hold for χ ∈ Ωlj , j < i. We use this induction hypothesis together with the induction hypothesis on the forms of commutators of α and β with γ ∈ [α, β] ∩ ∆b (the outer induction hypothesis) and

χ ∈ Ωlj with j < i (the inner induction hypothesis) to conclude the following. α,β Let ϕχ(s, x) = ρχ (s, 1, x) be deﬁned as before, and notice that (15.3) now holds only when a ∈ ker β, since we lack the linearity obtained in the base-ﬁber relations. While the proof of the following requires checking some complicated details, the following lemma follows from two simple ideas: splitting a commutator into a sum of two commutators requires a conjugation and reordering, and with careful bookkeeping, the reordering and conjugation can be shown to contribute polynomial terms only.

α Lemma 15.17. ϕχ(s1 + s2, x) = ϕχ(s1, x) + ϕχ(s2, ηs1 x) + s2p(s1, s2) for some polynomial p whose coeﬃcients are independent of x. α,β Proof. We assume that χ ∈ Ωli . Recall that ϕχ(s, x) = ρχ (s, 1, x) is the length of the χ-component of the unique path ρα,β(s, 1, x), written in circular ordering of the weights in D(α, β), which connects [s(α), 1(β)] · x and x. Notice that using only the free group relations, we get that:

(α) (β) (β) (α) (β) (α) [(s1 + s2) , t ] = (−t) ∗ (−s1 − s2) ∗ t ∗ (s1 + s2) (β) (α) (β) (α) = (−t) ∗ (−s1 − s2) ∗ t ∗ s2 (β) (α) (β) (β) (α) (β) (α) ∗ ((−t) ∗ s1 ∗ t ) ∗ ((−t) ∗ (−s1) ∗ t ) ∗ s1 (β) (α) (β) (β) (α) (β) (α) = (−t) ∗ (−s1) ∗ t ∗ (−t) ∗ (−s2) ∗ t ∗ s2 ∗ (β) (α) (β) (α) (β) ∗ (−t) ∗ s1 ∗ t ∗ [s1 , t ] (β) (α) (β) (α) (β) (β) (α) (β) (α) (β) = (−t) ∗ (−s1) ∗ t ∗ [s2 , t ] ∗ (−t) ∗ s1 ∗ t ∗ [s1 , t ] (α) (β) Figure6 illustrates the equality of [(s1 + s2) , t ] and the last term of the string of equalities above. We may replace the last line with

(β) (α) (β)−1 α,β −1 (β) (α) (β) α,β −1 (−t) ∗ s1 ∗ t ∗ ρ (s2, t, y) ∗ (−t) ∗ s1 ∗ t ∗ ρ (s1, t, x) α where y = ηs1 (x). Therefore, modulo commutator relations,

α,β α,β (β) (α) (β)−1 α,β (β) (α) (β) (15.7) ρ (s1+s2, t, x) = ρ (s1, t, x)∗ (−t) ∗s1 ∗t ∗ρ (s2, t, y)∗ (−t) ∗s1 ∗t . We now use the induction hypothesis. Notice that we would like to compute the χ term, so we must try to use the group relations known to put the expression into its canonical circular ordering. α,β In our computations, we will carefully reorder the weights appearing in ρ (s1 + s2, t, x), which in each case may introduce polynomial expressions in the χ term. This does not affect our result because we only wish to obtain the cocycle property up to a polynomial term. We first make certain 93 Figure 6. The cocycle-like property geometric arguments which correspond to the formal proof manipulating expression (15.7) which follows. Let us summarize the figure. Notice that in Figure6, there are five curves consisting of up to 5 colors each representing the (conjugates of the) ρα,β-terms in (15.7). The purple legs consist of weights in [α, β] ∩ ∆b, the blue legs consist of weights from the collection γ ∈ Ωlj with j < i, the red legs correspond to the weight χ ∈ Ωli (which we isolate in the current inductive step), the green legs consist of the other weights in Ωli , together with any weights of Ωlj , j > i, and the brown legs consist of the weights in [α, [α, β]] \ [α, β] (while this set will be empty by the Baker-Campbell-Hausdorff formula once the we conclude that the action is homogeneous, we cannot assume this now). We make some observations to justify the picture. First, since we know the base-fiber relations from Section 15.3 take polynomial forms independent of x, we may reorder the usual circular ordering expressing each ρα,β term to have all base weights (purple legs) appearing first. Since the fiber is abelian, we may express the blue, red and green legs in whichever order we wish without changing α,β the χ term. For each blue leg, we know that ργ take the form of Claim 15.6 by the induction hypothesis. Note that in the case of Ωl0 , there are no blue legs, so one may consider this to be the base case, as we do not require any inductive hypothesis. We elect to write the proof this way as the proof of the base and inductive steps are virtually identical. α,β (β) (α) Conjugating the ρ (s2, t, y) in (15.7) corresponds to “sliding” it along the t , s1 and in the opposite direction t(β) legs. Notice that all terms, with the possible exception of the purple and brown base legs, commute with one another, since the fiber is abelian. If γ is a green leg, then α,γ β,γ ρχ = ρχ ≡ 0 by Lemma 15.16, since χ ∈ Ωli and the green legs are in Ωlj with j ≥ i. Therefore, the nonlinear parts of the χ terms come from commuting the purple legs ([α, β] ∩ ∆b), blue legs 94 (Ωlj , j < i), and brown legs. The χ-contributions from commuting with the blue and purple legs are known to have polynomial form either by the arguments of Section 15.3 or by the outer induction, respectively. The new brown color curves correspond to the possible weights λ ∈ [α, γ] or [β, γ], where γ ∈ [α, β]. If γ ∈ ∆f , γ appears in some Ωlj , and we may use Lemma 15.16 if j ≥ i or the induction hypothesis if j < i to give a polynomial form of λ, which guarantees a polynomial contribution to χ by Section 15.3. If γ ∈ [α, β] ∩ ∆b, we claim that the λ terms take polynomial form. In the case when λ ∈ ∆f , this can be concluded from the outer induction on #(D(α, β) ∩ ∆b), since #(D(α, γ) ∩ ∆b), #(D(γ, β) ∩ ∆b) < #(D(α, β) ∩ ∆b). If λ ∈ ∆b, this is immediate from the homogeneous structure of XE0 . In summary, at each conjugating step when α,β conjugating ρ (s2, t, y), we obtain polynomial forms for all legs which feed into χ, yielding the desired cocycle-like property. α,β We also provide a formal, algebraic argument, using (15.7). We must commute the ρ (s2, t, x) (β) (α) (β) term with the t ∗s1 ∗(−t) term. By the outer induction, if γ ∈ [α, β]∩∆b, since #(D(α, γ)∩ (β) ∆b), #(D(γ, β) ∩ ∆b) < #(D(α, β) ∩ ∆b), we know that commuting (−t) past any such γ gives polynomials, possibly in other base weights or fiber weights, but all of which lie strictly between α and β. 0 0 0 0 Now suppose that χ ∈ D(α, β) ∩ ∆f satisfies χ ∈ [α, χ ] or χ ∈ [β, χ ], so that χ ∈ Ωlj , j < i by Lemma 15.16. Rearranging with the fiber weights, we know also by the inner induction and Section 15.3 that χ contributions from such rearrangements are polynomial form. Thus, commuting (−t)(β) α,β past each leg of ρ (s2, t, x) in the fiber also has χ-term which is a polynomial in s2 and t. We may (α) (β) similarly pass s1 and t through to arrive at a product of terms such that every contribution 0 from some γ ∈ [α, β]∩∆b or χ ∈ Ωlj with j < i is a polynomial. Any χ legs obtained by commuting polynomial terms with α or β are therefore also polynomial by Section 15.3, and these are the only α,β possible weights that can feed into χ by Lemma 15.16. The χ terms directly from ρ (si, t, x) are therefore the only nonpolynomial terms which could appear after putting the weights in a circular α,β α,β ordering, and must coincide with the original χ terms ρ (s1, t, x) and ρ (s2, t, y). α Thus we have shown that ϕχ(s1 +s2, x) = ϕχ(s1, x)+ϕχ(s2, ηs1 x)+P (s1, s2) for some polynomial P . Divisibility of P by s2 follows from taking s2 = 0 in this equality, giving ϕχ(s1, x) = ϕχ(s1, x) + P (s1, 0), so P (s1, 0) = 0 for every s1. This occurs if and only if P is divisible by s2. By Lemma 12.16, for every χ ∈ [α, β], either u ≥ 1 or v ≥ 1 (or both). Without loss of generality assume that u ≥ 1.

u Corollary 15.18. ϕχ(s, x) = cs for some c ∈ R which is either independent of x, or constant along the ﬁbers of a circle factor on which ker β has dense orbits.

E ∂ Proof. We claim that for every x ∈ X and Lebesgue almost every s1 ∈ , ϕ(s, x) exists. R ∂s s=s1 It suﬃces to show that ϕ(s, x) is locally Lipschitz in s. By Lemma 15.17,

α |ϕ(s, x) − ϕ(s1, x)| = ϕ(s − s1, ηs1 x) + (s − s1) · p(s, s − s1) u α ≤ |s − s1| ϕ(1, a · ηs1 x) + |s − s1| · |p(s, s − s1)| for a suitable choice of a ∈ ker β (using (15.3)). Since p is a polynomial and u ≥ 1, ϕ(s, x) has a Lipschitz constant in any neighborhood of s. Therefore, for almost every s1 ∈ R, ϕ is diﬀerentiable E α in s at s1. By Lemma 15.17, for every x ∈ X , Lebesgue almost every y ∈ W (x) has ϕ(s, y) ∂ E diﬀerentiable at 0. Therefore, f(x) = ∂s s=0 ϕ(s, x) exists on a dense subset of X , since it holds 95 α at Lebesgue almost every point of every leaf W (x). We claim that |f(x)| ≤ B for some B ∈ R whenever it exists. Indeed:

1 1 u |f(x)| = lim |ϕ(s, x)| = lim s |ϕ(1, as · x)| ≤ sup |ϕ(1, y)| s→0 s s→0 s y∈X where as ∈ ker β is chosen appropriately (using (15.3)), since u ≥ 1. Notice that if a ∈ ker β, then

∂ ∂ uα(a) −α(a) (u−1)α(a) f(a · x) = ϕ(s, a · x) = e ϕ(e s, x) = e f(x). ∂s s=0 ∂s s=0 Therefore, either u = 1 or f ≡ 0, since otherwise one may apply an element a with α(a) arbitrarily large to contradict the boundedness of f. Since u = 1 or f ≡ 0, f is constant along ker β orbits. α α We claim that f is also constant along W leaves whenever it exists. Indeed, if x2 ∈ W (x1) and f exists at both x1 and x2, choose ak ∈ ker β such that α(ak) → −∞ and ak · x1 → z for some z ∈ X (ﬁrst choose any such sequence, then choose a convergent subsequence). Since α is contracted by ak, ak · x2 → z as well. Then: 1 −α(ak) α(ak) f(xi) = lim ϕ(s, xi) = lim e ϕ(e , xi) = lim ϕ(1, ak · xi) = ϕ(1, z). s→0 s k→∞ k→∞ α Let mx denote the common value for f along W (x). Therefore, if f(s1) exists,

∂ ∂ (15.8) ϕ(s, x) = ϕ(s + s1, x) ∂s ∂s s=s1 s=0

∂ α = ϕ(s1, x) + ϕ(s, ηs1 x) + (s − s1)p(s1, s) = mx + q(s1) ∂s s=0 where q is some polynomial independent of x. Since ϕ(0, x) = 0, one may integrate mx + q(s1) to a get a polynomial form for ϕ at each x. This implies that f exists everywhere and varies continuously. Since f is invariant under ker β, if ker β has a dense orbit, f is constant, and hence mx is independent of x. Otherwise, ker β has an orbit dense in the ﬁber of some circle factor. Finally, u ϕ(s, x) = cs , since by (15.3), it must be u-homogeneous.

Proof of Claims 15.5 and 15.6. Since ϕχ takes a polynomial form, we may analyze the function α,β t 7→ ρχ (s, t, x) in the same manner as directly after Lemma 15.12 (which has the same conclusion k as Corollary 15.18). Indeed, if s, t ∈ R and a ∈ R satisﬁes α(a) = 0 and β(a) = log t, then by Corollary 15.18 and (12.4),

α,β v log t α,β v (15.9) ρχ (s, t, x) = e ρχ (s, 1, a · x) = t ϕχ(s, a · x) v u u v α,β u α,β = ca·xt s = s t ρχ (1, 1, a · x) = s ρ (1, t, x). k Therefore, for any s, t ∈ R and a ∈ R ,

uα(a)+vβ(a) α,β α,β α(a) β(a) uα(a) α,β β(a) (15.10) e ρχ (1, t, x) = ρχ (e , e t, a · x) = e ρχ (1, e t, a · x), where the first equality follows from (12.4) and the second follows from (15.9). Then (15.10) is analogous to (15.3) which is the principal ingredient tool in studying properties of ϕ. Then, as in the proofs of Lemma 15.13 and Corollary 15.14, the weights of D(α, β) form a Lie group action, α,β and by Lemma 12.15 we get the desired polynomial form for ρχ (s, t, x). 96 Case: 2-step nilpotent. As in the base-fiber case, there cannot be negatively proportional weights appearing in D(α, β), so the corresponding fiber weights all commute, and the argument in the abelian case works verbatim in the case of a 2-step nilpotent group.

16. Proofs of Theorems 1.7 and 12.19 k k ` We prove all results for R -actions, since given an R × Z action, one may build a suspension k+` k ` ` space X˜ with an R action into which the R × Z action embeds, with a T factor. This action has a non-Kronecker rank one factor if and only if the original action has a non-Kronecker rank one ` factor by Lemma 3.13. The resulting homogeneous action will have T -factor, and the fibers of this k ` factor will be conjugate to the original R × Z action. k Proof of Theorems 1.7 and 12.19. Let R y X be a Cartan action satisfying the assumptions of Theorem 1.7 or 12.19, and ∆ denote its set of weights. Then Section 13 allows us to construct a sequence of subsets ∅ ( E1 ( ··· ( En = ∆ such that each Ei is an ideal, and there are no ideals between Ei and Ei+1 for i = 1, . . . , n − 1. Then En−1 is a maximal ideal of ∆, and XEn−1 has constant pairwise cycle structures by Lemma 13.14. Hence by Theorem 14.2, XEn−1 is a homogeneous space and the induced Cartan action is homogeneous. Inductively, we will show that if the induced action on XEj is homogeneous, then so is the induced action on XEj−1 . Notice that these are exactly the assumptions laid out at the start of Section 15. Then Propositions 12.12, 15.6, 15.8 and 15.15 imply that the action on XEj−1 has constant pairwise cycle structures. Hence by Theorem 14.2, the induced action on XEj−1 is homogeneous. Therefore, after finitely many steps, we conclude that the action on X∅ = X is homogeneous. This finishes the proof of Theorem 12.19. In the case of Theorem 1.7, we will first show that the conjugating map h : X → G/Γ is C1,θ for α some θ ∈ (0, 1). For any α ∈ ∆, h intertwines the actions of the one parameter groups ηt as well k β as the R actions on X and G/Γ by Proposition 13.10. Notice that dh maps the generator of ηt on X to the one on G/Γ. The latter is smooth and the first is Hölder since W α is a C1,θ0 foliation for some θ0 and the generator is chosen to be unit length according to the Hölder metric from Section 1,θ0 β 2 k 5.6. In consequence, h is C along all ηt orbits. Similarly, h is C along the R orbit foliation. 2 ∞ β We in fact claim that h is as smooth as the action (C or C ) along ηt orbits. We let r = 2 or ∞ depending on the version of the theorem. Fix a Lyapunov exponent α and some smooth Riemannian metric on X (not necessarily the one of Section 5.6). Let ϕa : X → R be defined to be the norm derivative of a restricted to Eα with respect to the chosen Riemannian metric. That is, ϕa(x) = ||da|Eα (x)|| and ϕa is Hölder. Notice that if the ratio of the Riemannian norm to the dynamical norm of Section 5.6 is given by ψ(x), then ψ(x) is Hölder (since the dynamical metric is Hölder) and

α(a) −1 (16.1) ϕa(x) = e ψ(a · x)ψ(x) . Take L = X × R to be the trivial line bundle over the manifold X, and let π(x, t) be the point k which has (signed) distance t from x in the smooth Riemannian metric on X. For each a ∈ R , there exists a unique lift a˜ to L such that π(ã(x, t)) = a · π(x, t). Notice that since each fiber corresponds to a Cr submanifold, a˜ is a Cr extension as in Section 3.3. Let H : L → L be the map H(x, t) = (x, H (t)), with H uniquely satisfying ηα (x) = π(x, t). Then observe that x x Hx(t) HaH˜ −1(x, t) = (a · x, eα(a)t) by (12.1). Notice that the derivative of π(x, t) with respect to t is the unit vector of Eα with respect to the smooth Riemannian metric, and the derivative of ηα (x) is the derivative of H times the unit Hx(t) x 97 α 0 1 vector of E with respect to the dynamical norm. Therefore, Hx(t) = ψ(π(x, t)), so Hx ∈ C (R, R) varies continuously with x. Now, finally we modify H slightly by defining G(x, t) = H(x, ψ(x)−1t). 0 r Then Gx(0) = 1 and G is still a linearization by (16.1). Therefore, G is a system of C normal form r r α coordinates by Theorem 3.10(b), and Hx(t) is C in t. This implies that h is C along W -leaves. We recall the main result of Journé in [36]: Given two continuous transverse foliations F1 and F2 1,θ with uniformly smooth leaves on a manifold M. Suppose f is a function uniformly C along F1 1,θ ∞ and F2. Then f is C . Note that the theorem in [36] states a version for C functions along F1 1,θ and F2. The proof however works for uniformly C functions as the author explicitly states. k Recall that a0 is the regular element in R which determines the positive and negative weights ∆+ and ∆−. Order the weights λ ∈ ∆− cyclically: λ1, . . . λl. Then inductively we see from Journé’s 1,θ λ1 λ1 λ2 λ theorem that h is C along the foliations tangent to E ,E ⊕ E ,..., ⊕I=1,...,lE i . Clearly, the s 1,θ s last foliation is nothing but the stable foliation of a0, Wa0 . Hence, h is C along Wa0 , and similarly u 1,θ along the unstable foliation Wa0 of a0. Using Journé’s theorem again, we get that π is C along the weak stable foliation, by combining stable and orbit foliations, and then on X combining weak stable and unstable foliations. In the C∞ setting, one may repeat the arguments with the C∞ version of the regularity lemma ∞ [36] to obtain that h is C .

Part III. Structure of General Totally Cartan Actions In Part III, we return to a general totally Cartans action, and prove a structure theorem for them. In particular, since we allow rank one factors, we no longer have the Hölder metrics from b) of the higher rank assumptions. Instead, we will again rely on Lemma 5.4, which only requires the totally Cartan assumption to hold. From Parts I and II, we have established a dichotomy: either there exists a non-Kronecker rank one factor for the action which is an Anosov 3-ﬂow, or the action is homogeneous. The key insight in obtaining Theorem 1.11 is the following: any rank one factor of the action actually splits as a direct k−1 product, and the R action on the complementary manifold is still totally Cartan (Proposition 19.10). This allows us to “peel oﬀ” rank one factors until there are none left, and we may apply the rigidity results of Part II to get that the action must (covered by) the direct product of its rank one factors, and some homogeneous action.

17. The Starkov component r k T Fix a C , totally Cartan R action on a manifold X, and let S = β∈∆ ker β denote its Starkov component (recall the deﬁnition made directly before Remark 1.9). Proposition 17.1. The action of S factors through a torus action.

Proof. Fix some Riemannian metric on X. Recall that by Lemma 5.4, if a ∈ ker β, then ||da|Eβ || is uniformly bounded above and below by constants L and 1/L, respectively. Since a ∈ S, we have this for every coarse exponent β and since da|T O is isometric (where O is the orbit foliation of the k R action), the action of S is equicontinuous. By the Arzelà-Ascoli theorem, the closure of S in the group of homeomorphisms of X is a compact group, call it H. Observe that if h ∈ H, then h is an equicontinuous homeomorphism that commutes with the k k n n n n R -action. Therefore, since for every a ∈ R , d(a · h(x), a · x) = d(h(a · x), a · x) is uniformly k bounded above and below, if h is close to the identity in H, h(x) ∈ R · x. Therefore, h determines k a continuous function τ : X → R by h(x) = τ(x)x. A simple computation using the fact that 98 k k h commutes with the R -action shows that τ is constant on R -orbits and since there is a dense k R -orbit, τ is constant everywhere. Therefore, the action of a neighborhood of the identity in H coincides with a subgroup of the k R -action. Since H is the closure of a connected group, H is connected, and the action of H and k the action of a subspace of R coincide. But any a 6∈ S cannot act equicontinuously (since then β(a) 6= 0 for some β), so we know that S factors through H. Since H is a compact factor of S, ` H = T for some ` and we obtain the result. Lemma 17.2. The quotient X/S is a manifold.

k−dim(S) k Recall the X/S is called the Starkov factor of X. It is clear that any transverse R ⊂ R descends to a locally free action and the coarse Lyapunov foliations W β descend for every β ∈ ∆, giving X¯ = X/S a transitive, totally Cartan action. Proof. The proof is identical to Proposition 7.6, we briefly summarize the method. We must show that StabS(x) is independent of x. To this end, we show that for every x, y ∈ X, StabS(x) ⊂ StabS(y). Then reversing the roles of x and y shows equality. It suffices to show that StabS(x) ⊂ k β k StabS(y) for every y ∈ R ·x and W (x) for any β ∈ ∆. The proof for R is trivial by commutativity. One may use the preservation of the β leaves together with normal forms to show that if a ∈ S and a · x = x, then a|W β (x) = Id. 18. Centralizer Structure k 18.1. Structure of a general centralizer of a totally Cartan Action. Let R y X be a tran- k sitive, totally Cartan action with trivial Starkov component. Fix an R -periodic orbit p. Consider |∆| V = R as a vector space, whose entries represent the logarithms of eigenvalues of the derivatives β1 β|∆| k of a map along the foliations W ,...,W at p. Let Λp ⊂ ZDiff1 (R y X) denote the subgroup of k the centralizer of the R action consisting of elements fixing p. Define a homomorphism ψ :Λp → V by setting ψ(λ) = log λ | β (p),..., log λ | β (p) . ∗ W 1 ∗ W |∆| Lemma 18.1. The homomorphism ψ is injective.

Proof. The proof is very similar to that of Proposition 7.3. Suppose that ψ(λ) = 0. Then λ∗(p) = Id. Since λ acts by linear transformations in the normal forms coordinates on the W β-leaves, λ also fixes the leaves W β(p) pointwise for every β ∈ ∆. We may inductively show that λ also fixes k the R -saturated accessibility class of p, which is all of X. Therefore, λ = Id and the map is injective. k The following lemma is immediate by jointly diagonalizing the derivatives of elements a ∈ R such that a·q = q to obtain coefficients of the Lyapunov functionals, and pushing the metric around k with the desired rescaling along its R -orbit. The procedure is analogous to the metric constructed when there is a circle factor in the proof of Theorem 2.2 in Section 11. k Lemma 18.2. If q is an R -periodic point in X and β is a weight, there exists some cq ∈ R+ β k cqβ(b) k a norm ||·||β = ||·||β,aq on Eaq, a ∈ R such that ||b∗(v)||β = e ||v||β for every b ∈ R and β k v ∈ Eaq, a ∈ R . k Let F be any finite collection of periodic points in X with distinct R -periodic points, one of F which is p. Let Λp ⊂ Λp be the subgroup of Λp defined by:

F n k k o Λp = f ∈ Λp : f(R · q) = R · q for every q ∈ F . 99 F Then Λp has finite index in Λp, since for each q ∈ F , there are only finitely many other periodic k F |F |+1 orbits with the same R -stabilizer. We now define a new homomorphism Ψ:Λp → V . To p, and each point in F , we associate an element of V . For p, we use ψ. For q ∈ F , associate the element log ||λ∗| β (q)|| ,..., log λ∗| β (q) . W 1 β1 W |∆| β|∆|

Proposition 18.3. ΨF is an injective homomorphism. Proof. That it is injective follows from Lemma 18.1. To see that Ψ is a homomorphism, suppose F that f, g ∈ Λp . It suﬃces to show that Ψ is a homomorphism in each of the V -components. Fix q ∈ F and β ∈ ∆. Then:

(fg)∗|W β (q) = f∗|W β (g(q)) · g∗|W β (q). Upon taking norms and logarithms, this proves that Ψ is a homomorphism, provided that we F F show that f∗|W β (g(q)) = f∗|W β (q) for every f, g ∈ Λp . Notice that since g ∈ Λp , it suffices to k show that ||f∗|W β ||β is constant along the R orbit of q. To see this, notice that since fa = af, cqβ(a) cqβ(a) (fa)∗|W β = (af)∗|W β . Therefore, ||f∗|W β (aq)||β · e = e · ||f∗|W β (q)||β, and ||f∗|W β (·)||β is k constant along the R -orbit of q as claimed. We now fix a choice of F . For each weight β, find an element a such that W β is the slow foliation for a (this is possible by Lemma 8.4), and again let Wdβ denote the complementary foliation within u Wa (x) for a suitable choice of a. Since periodic points are dense by Theorem 5.13, for each β ∈ ∆, we may pick another periodic point q very close to some point x ∈ W β(p) distinct from p at distance ε0 > 0 from p (so that both q and x lie in some chart around p in which local product structure holds). cs u By local product structure,there is a unique local transverse intersection Wloc(q)∩Wloc(p) = {y} and β β 0 since d(p, x) = ε0, by choosing q sufficiently close to x, we may assume that Wd(y) ∩ W (p) = {x } has x0 6= p. (again, see Lemma 8.4). Pick points x and q for every β ∈ ∆, and let F denote the collection of such points q. F |F |+1 Lemma 18.4. ΨF (Λp ) is a discrete subgroup of V . F |F |+1 Proof. Suppose that λ ∈ Λp is very close to 0 ∈ V , so that ||λ∗|W γ (q)||γ is very close to 1 for every γ and at every q ∈ F . cs By construction, for λ such that Ψ(λ) is sufficiently close to 0, λ fixes the intersection of Wloc(q)∩ u β 0 0 Wloc(p) = {y}, since the intersection is locally unique. Then y ∈ Wd(x ) for some x 6= p, and since Wdβ(y) and W β(p) are both invariant and have derivatives very close to one, their local intersection is also unique and preserved. But then λ(x0) = x0, so λ fixes two points on the same β leaf. This implies that λ fixes the entire β leaf since it must be linear in the normal forms coordinates. So λ∗|W β (p) = 1. Since this can be arranged for each β, we get that ψ(λ) = 0 and hence by Lemma F |F |+1 18.1, we conclude that λ = Id. Therefore, Ψ(Λp ) is discrete in V . k F k ∼ k F F k Since p ∈ F is an R -periodic point, Λp ∩ R = Z . Therefore, Λ := Λp · R is a finite index subgroup of Λ, and is exactly the set of elements which fix the orbits of elements of F , not necessarily fixing p. F Lemma 18.5. ΨF extends to an injective homomorphism from Λ . F k ` F ∼ k ` Proof. Notice that Λ is an abelian group isomorphic to R × Z for some `, and that Λp = Z × Z is a lattice in ΛF . Since Ψ is now an injective homomorphism from a lattice to a lattice, it must extend to an injective homomorphism on the corresponding vector spaces. 100 18.2. The totally Cartan property of actions by centralizers. To understand smooth centralizers, we first show an easier property, that the C1-centralizer actually has higher regularity. k r Lemma 18.6. Let R y X be a C transitive Cartan action, with r = 2 or r = ∞ and suppose k k 0 that f ∈ ZDiff1 (R y X). Then f ∈ ZDiffr (R y X), where r = (1, θ) or ∞, respectively. Proof. Fix a coarse Lyapunov exponent β ∈ ∆. By Theorem 3.10, there exists a Cr family of β β 1 β normal forms charts ψx : R → W (x), such that any C -diffeomorphism preserving the leaf W k β and commuting with the R -action must be linear in the coordinates provided by ψx . Therefore, k r β any f ∈ ZDiff1 (R y X) must be uniformly C along the leaves of W . Since each stable manifold can be built by adding coarse Lyapunov exponents, one at a time, along a circular ordering, one can iteratively apply Journé’s theorem [36] to get that f is uniformly Cr0 along the stable and unstable foliation of a fixed element a. Therefore, since the stable and weak unstable are complementary and uniformly transverse, we may conclude that f is uniformly Cr0 by another application of Journé’s theorem. The following is crucial in the proof of Theorem 1.11: it allows us to produce a totally Cartan action which is virtually self-centralizing from one which is not. The main difficulty is showing that any new elements added are still Anosov. k Theorem 18.7. If R y X is a transitive, totally Cartan action with trivial Starkov component, k l k 0 there exists a finite index subgroup R × Z ⊂ ZDiffr (R y X) such that the suspended action k+` R y X˜ is a transitive, virtually self-centralizing, totally Cartan action with trivial Starkov component. k r0 Proof. Note that by Lemma 18.6, we may assume that any element f ∈ ZDiff1 (R y X) is C . We k F k ` take the subgroup R × Λp , which is isomorphic to R × Z for some ` by Proposition 18.3 and k ` Lemma 18.4. Since there are only finitely many points with a fixed period, it is clear that R × Z k ˜ k ` is finite index in ZDiff1 (R y X). Let X be the suspension space of R × Z y X, as described in Section 3.5. Fix a weight β ∈ ∆ with coarse Lyapunov foliation W β. Recall that β is not yet a linear k functional, but we can make sense of “ker β”, which is a codimension one hyperplane in R which is the complement of the elements which either expand or contract W β. We will show that there k+` k are linear functionals β˜ : R → R such that if a 6∈ ker β˜ ⊂ R , then a either uniformly expands or uniformly contracts all leaves of W β ⊂ X˜. Notice the original hyperplane ker β will not suffice since k k it will be codimension ` + 1 in R + `. Still, our construction will imply that R ∩ ker β˜ = ker β. There are two cases for each β ∈ ∆: either ker β has a dense orbit in X (or ker β is dense in the k fiber of some circle factor in X), or there is a non-Kronecker rank one factor of R y X onto which W β projects as the stable foliation. If ker β has a dense orbit (or ker β is dense in the fiber of some circle factor), then by Theorem 2.1 and the proof of Theorem 2.2 in Section 11, there exists a Hölder metric ||·||β and a linear functional k β β (which we abusively denote by β : R → R) on E such that ||a∗|Eβ (v)||β = e (a)v β. The metric k ` is unique up to global scalar multiple. If f ∈ R × Z , then f∗ ||·||β is also a norm satisfying the β same property. Therefore, if v ∈ E is any unit vector, the map f 7→ ||df|Eβ (v)|| is linear, and we k+` denote its linear extension to R by β˜. It obviously satisfies the desired properties. β Now assume that W projects as the stable foliation in some rank one factor ft y Y , with k projection p : X → Y and linear functional σ : R → R such that p(a · x) = fσ(a)p(x). Note that k (ft) must be transitive since the R -action is transitive. Let ν denote the Margulis measure for ft (as in the proof of Theorem 3.12), and ν denote the family of conditional measures provided 101 by Lemma 3.11 satisfying (3.1). Since the foliations W β map injectively onto W u(y), one can lift β β β β(a) β the measures to get a family of measures νx on W (x) ⊂ X such that a∗νx = e νa·x, where β(a) = hσ(a). k ` β Finally, if b ∈ R ×Z , then b∗νx is a continuously varying family of measures satisfying conditions ˜ k ` β (1)-(3) of Lemma 3.11. Therefore, there is a linear functional β : R × Z → R such that b∗νx = β˜(b) β ˜ k+` e νb·x. The extension of β to R then satisfies the desired properties, since once b expands the β lifted Margulis measures νx , its normal forms must also be expanding (similarly for contracting). So we have shown that the suspended action is totally Cartan. It is transitive since it is the suspension of a transitive action. It has trivial Starkov component by Lemma 18.5 (no element k+` of R is in ker β˜ for every β˜). Finally, we must show that the suspended action is virtually k+` self-centralizing. Suppose that f : X˜ → X˜ commutes with the R action. Since the action is a ` suspension, the action factors over a transitive torus flow on T . The fibers are all diffeomorphic to β k X, and are saturated by the foliations W and R -orbits. Since f must preserve these foliations, ` f maps fibers to fibers and descends to a map of T commuting with every translation. Therefore, ` the action of f is a translation, and we may choose b ∈ R such that f ◦ b fixes every fiber and k 0 commutes with the R -action. Therefore, f ◦ b = b · γ, where γ is chosen from a finite collection of k k ` 0 k ` −1 0 k+` representatives of ZDiff1 (R y X)/(R × Z ) and b ∈ R × Z . Hence, f = b b · γ, and R is k+` ˜ k+` ˜ finite index in ZDiff1 (R y X). That is, R y X is virtually self-centralizing.

19. Virtually Self-Centralizing Actions Before analyzing the structure of totally Cartan actions, we ﬁrst consider the smoothness properties of centralizer actions for the models we wish to conjugate to, culminating in Proposition 19.5.

k1+k2 0 Lemma 19.1. If R y X = Y1 × · · · × Yk1 × X is the direct product action of Anosov ﬂows on k2 0 3-manifolds Y1,...,Yk1 and a homogeneous action R y X , then any homeomorphism f : X → X 0 centralizing the action can be written as f = g1 × g2 × . . . gk1 × f , where each gi centralizes the 0 0 Anosov ﬂow on Yi and f centralizes the homogeneous action on X .

Proof. It suffices to show that f takes sets of the form {y1} × · · · × Yi × · · · × {yk1 } × {x} and 0 {y1} × · · · × {yk1 } × X to sets of the same respective form. From this, it is easy to see how to define 0 gi and f , and show that f is actually equal to their direct product. Notice that f preserves the coarse Lyapunov foliations, since they are intersections of stable manifolds, which are defined through dynamical properties preserved by f by commutativity. It follows immediately that f has the property discussed above.

Lemma 19.2. Let (gt) be a transitive Anosov flow on a 3-manifold Y . Then ZDiff1 (gt) is finite index in ZHomeo(gt).

Proof. This is immediate from Theorem 3.12.

Remark 19.3. It is not always true that ZDiff1 (gt) = ZHomeo(gt). Indeed, suppose that A is an 2 2 2 Anosov automorphism of T such that there exists a commuting translation T : T → T . Choose a fixed point of the automorphism p, to that T (p) is another fixed point. Perturb the automorphism to a diffeomorphism f which still fixes p and T (p), but so that near the fixed point p, f∗(p) 6= f∗(T (p)). Then by structural stability, there is still a homeomorphism T˜ commuting with f such that T˜(p) = p. But, since f∗ does not agree at p and T˜(p), T˜ cannot be differentiable at p. One can obtain a flow version by suspending, or repeating this trick on a geodesic flow with a discrete symmetry. 102 k k Lemma 19.4. Let R y G1/Γ1 and R y G2/Γ2 be two Cartan homogeneous actions, and let ∞ f : G1/Γ1 → G2/Γ2 be a homeomorphism which intertwines the actions. Then f is C .

Proof. By Proposition 14.21, we may assume without loss of generality that each Γi is discrete. Let µi denote the Haar measure on Gi/Γi. Then f∗µ1 is also an invariant measure for the action on k G2/Γ2, and hµ1 (a) = hf∗µ1 (a) for every a ∈ R . When a is Anosov and transitive, µi is a measure of maximal entropy for a, and this measure is unique [10, Theorem 5.2(2)]. Since topological entropy is a homeomorphism invariant, f∗µ1 = µ2. Fix a coarse Lyapunov exponent α, and let U be a one-parameter subgroup of G1 whose orbits α U are the leaves of the foliation W . Then µ1 has a canonical disintegration along U, µx , which is the Haar measure on U. This can also be obtained using a measurable partition subordinate to W α. Using the dynamics to expand the atoms of the partition, one gets that Haar measure on U-orbits can be obtained as the unique conditional measure which is normalized so that the ball of radius 1 in U has measure 1. This procedure is standard, more details can be found, for instance, in [5, Section 3.3] or [17, Proposition 3.2]. s n Since stable manifolds are defined using a topological condition (x ∈ Wa (y) if and only if d(a · n x, a · y) → 0), f takes stable manifolds for the action on G1/Γ1 to stable manifolds for the action α α on G2/Γ2. Each W is obtained as an intersection of stable manifolds, so f maps the foliation W α on G1/Γ1 to a foliation W¯ on G2/Γ2. The foliation is also a coarse Lyapunov foliation, and also U¯ U U¯ have corresponding conditional Haar measures µ¯x . Since f∗µ1 = µ2, f∗µx = λxµ¯f(x) for µ1-almost every x ∈ G1/Γ1 since the disintegrations are defined uniquely up to scalar (recall that we obtained U fixed families by normalizing the unit ball to have µx -measure 1). k c α(a) Now, the derivative of the R -action is given by e i for some ci ∈ R+ along coarse Lyapunov (c −c )α(a) foliations for both the action on G1 and action on G2. Therefore λx = e 1 2 λa·x. x 7→ λx is a measurable function, so it is continuous on compact sets of large measure by Lusin’s theorem. k Any homogenous Anosov R -action is ergodic with respect to Haar measure (this follows from a straightforward adaptation of the Hopf argument for Anosov flows). We may therefore choose a point of the Lusin set which returns to it with positive frequency by the ergodic theorem. Since the Lusin set is compact, λx must be bounded on it, which is a contradiction unless c1 = c2. Therefore, k λx is constant on R -orbits. This implies that λx is constant, and therefore the partial derivative of f along W α is constant and exists everywhere by continuity. k k Since f commutes with the R action, it is trivially differentiable when restricted to R -orbits. ∞ Therefore, by [45, Theorem 2.1], f is C .

k1+k2 0 Proposition 19.5. If R y X = Y1 × · · · × Yk1 × X is the direct product action of transitive k2 0 k1 k2 0 Anosov flows on 3-manifolds and a homogeneous action R y X , then R × ZDiff1 (R y X ) k1+k2 k1+k2 and ZDiff1 (R y X) are both finite index in ZHomeo(R y X). Proof. The proposition is immediate from Lemmas 19.1 through 19.4. k r Lemma 19.6. If R y X is a C action, then if the action is virtually self-centralizing, so is the action on any finite-to-one factor of the action. Proof. Let π : X → Y be a finite-to-one factor map onto which the action descends. Recall that a map f : Y → Y lifts to X if and only if f# leaves π#(π1(X, x)) invariant (where # denotes the induced homomorphism on fundamental groups). Since π is a finite-to-one cover, π#(π1(X, x)) k is a finite index subgroup of π1(Y, π(x)). Therefore, ZDiff1 (R y Y ) acts by automorphisms of π1(Y, π(x)), and hence the cosets of π#(π1(X, x)). In particular, a finite index subgroup of the k centralizer on Y lifts to X, and this group must contain the R action as a finite index subgroup. 103 k k Therefore, ZDiff1 (R y Y ) contains the R action as a finite index subgroup, and the factor action is virtually self-centralizing. r k 19.1. The Higher-Rank Factor. Fix a C totally Cartan action R y X with trivial Starkov component, with r = 2 or r = ∞. We assume for this section that π : X → Z is a C1,θ or C∞ k ` (if r = 2 or ∞, resp.) factor map, with homomorphism σ : R → R and a totally Cartan action ` β R y Z such that π(a · x) = σ(a) · π(x). Assume that every coarse Lyapunov foliation W either subfoliates the fibers of π, or has leaves uniformly transverse to the fibers of π. Let H denote ker σ, and for ease of notation, let r0 = (1, θ) or ∞ depending on whether r = 2 or ∞, respectively.

−1 r0 Lemma 19.7. If the action of H on π (z0) is C conjugate to a homogeneous action with no non- k Kronecker rank one factors, then there exists a finite extension R y X1 such that every element −1 k of a finite index subgroup of ZDiff1 (H y π (z0)) extends to an element of ZDiff1 (R y X1). k Furthermore, if every finite extension of R y X is virtually self-centralizing, then for every fiber π−1(z), the action of H is a virtually self-centralizing, totally Cartan action with trivial Starkov r0 −1 component. The action is C -conjugate to the homogeneous action over π (z0).

Proof. Let ∆f ⊂ ∆ denote the set of weights of the action which subfoliate the fibers of π, and ∆b ⊂ ∆ be the weights of the action which are transverse to the fibers of π. For each β ∈ ∆b, the β β foliations WX on X descend to foliations WZ on Z which are the corresponding coarse Lyapunov ` 0 β −1 foliations for the R -action there. If z ∈ WZ (z), we will produce homeomorphisms fz,z0 : π (z) → −1 0 −1 β π (z ) which intertwine the H := ker σ actions. Indeed, notice that if x ∈ π (z), then WX (x) β 0 covers WZ (z). Therefore, if z is sufficiently close to z, there exists a unique intersection point β −1 0 fz,z0 (x) := Wx (x) ∩ π (z ). It is easy to see that fz,z0 is well defined and intertwines the H-actions. Since the H-actions −1 over π (z0) do not have non-Kronecker rank one factors, ker γ has a dense orbit in the fiber for 0 β every γ ∈ ∆f (or is dense in the fiber of some circle factor). Since the fiber over every z ∈ WZ (z) has its H-action topologically conjugated by the map fz,z0 , the H ∩ ker γ action on the fibers over π−1(z) are dense, or dense in the fiber of some circle factor. By Corollary 12.18, the action also has subresonant Lyapunov coefficients, so by Theorem 1.7, the action over π−1(z) is Cr0 conjugate to r0 a homogeneous action. Therefore, by Lemma 19.4, the map fz,z0 is C . We let ρ denote a path in β ` Z in the foliations WZ with β ∈ ∆b, and R -orbit foliation, and Pz0 denote the set of such paths based at z0. −1 β Suppose that f ∈ ZDiff1 (H y π (z0)). If ρ is a path with only one leg along the WZ -foliation 0 −1 r0 1 ending at z , let Φ(ρ)(f) = fz,z0 ◦ f ◦ fz,z0 . Since the maps fz,z0 are C , Φ(ρ)(f) is C -conjugate to k −1 f. If a ∈ R , we similarly let aΦ(a)(f) = a ◦ f ◦ a . We can extend the definition of Φ to Pz0 by composing the conjugacies along individual legs. Since Φ is defined by conjugacies, it follows that

Φ(ρ)(f ◦ g) = Φ(ρ)(f) ◦ Φ(ρ)(g). Let Cz0 denote the set of cycles in Pz0 , so that the restriction of −1 Φ to Cz0 is an automorphism of ZDiﬀ1 (H y π (z0)). That is,

−1 Φ: Cz0 → Aut(ZDiff1 (H y π (z0))) is a well-defined group action. We claim that the image of Φ(Cz0 ) is a finite group. Indeed, fix F −1 a periodic orbit p and finite set F as in Lemma 18.4, so the group Λp ⊂ ZDiff1 (H y π (z)) F is the set of elements which fix p and fix the orbits of the points in F . Then Λp is finite index −1 in ZDiff1 (H y π (z)). Any automorphism acts on the space of cosets, so there is a finite index F F subgroup of Cz0 which preserves Λp . We will show that every element Λp extends to an element of the centralizer of some finite cover of X. 104 Now, (Φ(σ)(f))∗(q) must be conjugate to f∗(q) for any q ∈ F ∪{p}. So the map which sends f to the set of eigenvalues of f∗(p) along its coarse Lyapunov foliations is invariant under Φ(σ). Recall that the map Ψ from Proposition 18.3 is defined to be the ordered list of eigenvalues of f at the points of F ∪ {p}, and that Ψ is injective. Therefore, Φ(Cz0 ) has finite image.

This implies that ker Φ has finite index in Cz0 . Let us build a finite cover of Z and X. The cover of Z is constructed in the following way. Recall that if we quotient Pz0 by Cz0 , we recover Z. If instead we quotient by ker Φ ⊂ Cz, a finite index subgroup, we get a finite cover of Z, call it Z1. To build the cover of X, we let X˜1 denote the set of pairs (x, ρ), where x ∈ X and ρ is an element ˜ of Pz0 such that π(x) = e(ρ). Again, quotienting X1 by Cz0 recovers X, so quotienting by ker Φ gives a finite cover X1 = X˜1/ ker Φ. There are two canonical projections from X1, one onto Z1 (the second coordinate) and one onto X (the first coordinate), so that the following diagram commutes:

X1 X

Z1 Z

Notice that all actions and foliations lift to X1, and that by construction, the group Cz (when con- −1 structed on Z1 rather than Z) acts trivially on ZDiff1 (H y π (z0)). Therefore, if f ∈ ZDiff1 (H y −1 r0 ˜ k π (z0)), one may build a unique C diffeomorphism f on X1, commuting with the R action, −1 which restricts to f on π (z0) by letting f act on the fiber above a point z ∈ Z1 by Φ(ρ)f, where ˜ ρ ∈ Pz0 ends at z. Notice that f is continuously differentiable along the fibers by construction, β k that f˜ acts trivially on the W -foliations by construction, and that f˜ is smooth on the R -orbit k foliations since it commutes with the R action. Therefore, all partial derivatives of f˜ exist and 1 k ˜ vary continuously, so f is C . If R y X1 is virtually self-centralizing and f fixes p and elements ˜ of F , f, and hence f must be the identity by Proposition 18.3.

k r 19.2. Proof of Theorem 1.10 and Theorem 1.11. Let R y X be a C transitive totally Cartan action, with r = 2 or r = ∞. We first take steps to eliminate the Starkov factor, and control the centralizers on embedded actions. Then by Lemma 17.2, the Starkov factor of the action exists and the induced action on it has trivial Starkov component. By Theorem 1.7, either there is a factor of the action on the Starkov factor which is an Anosov flow on a 3-manifold, or the action r0 k−1 is C -conjugated to a homogeneous action. If there is such a factor, consider the R action on the fiber. Since the fiber embeds into a Cr action by construction, it has subresonant Lyapunov coefficients by Corollary 12.18. Thus, the action on the fiber has the same dichotomy. One may thus continue taking fibers of rank one actions within fibers until no non-Kronecker rank one factor can be taken, and the action on some fiber is Cr0 -conjugate to a (possibly trivial) homogeneous action. We therefore satisfy the assumptions of Lemma 19.7 (which follows trivially if the homogeneous action is trivial). Let X1 denote the finite cover of the Starkov factor for which every element of a finite index subgroup of the centralizer of the homogeneous action extends to a Cr0 diffeomorphism. Let X˜ denote the suspension space as constructed in Theorem 18.7. Note that by construction, the k+` R -action on X˜ satisfies the assumptions described at the start of Section 19.1, but also satisfies −1 that the action of H on π (z0) is virtually self-centralizing by Lemma 19.7. Furthermore, notice that if the action had trivial Starkov component and every finite extension of the action was virtually self-centralizing, we could take X˜ = X to have these properties. Since the action on X1 will embed k+` into the R -action on X˜ it suffices for both Theorem 1.10 and 1.11 to show that some finite cover of the action on X˜ is isomorphic to a direct product of Anosov flows and a homogeneous action. 105 k r We therefore let R y X be a C totally Cartan action with trivial Starkov component, for r = 2 or r = ∞. If the action does not have a rank one factor, by Theorem 1.7 (proved in Section 16), we conclude that the action is homogeneous. Otherwise, there exists a rank one factor π1 : X → Y0 with associated coarse Lyapunov exponent α0. Fix any y0 ∈ Y0, and let −1 Z1 = π1 (y0). Then the action of ker α0 on Z1 is a transitive totally Cartan action with trivial Starkov component. Iterating this procedure, we arrive at a chain of spaces Z1,Z2,...,Zn, 3- manifolds Y0,Y1,Y2,...,Yn−1 equipped with transitive Anosov flows and associated coarse Lypaunov 1,θ ∞ exponents αi, and C or C projections πi+1 : Zi → Yi such that: Ti (1) j=1 ker αj acts by a transitive, totally Cartan action with trivial Starkov component on Zi, and −1 (2) Zi+1 = πi+1(yi) for some yi ∈ Yi. We continue this process until we can no longer take rank one factors. That is, until the action on r0 Zn is a C transitive totally Cartan action with trivial Starkov component with no non-Kronecker rank one factors. We allow the possibility that Zn is a singleton. Notice that because it embeds into a C2 action, it has subresonant Lyapunov coefficients by Corollary 12.18. Therefore, the action 1,θ on Zn is C conjugated to a (possibly trivial) homogeneous action. By the discussion at the start of this section, we may assume without loss of generality that the the homogeneous action on the fibers is virtually self-centralizing. We wish to show that the action on some finite cover of X˜ is smoothly conjugated to the direct product of (some finite covers of) the homogeneous action on Zn and the Anosov flows on each Yi. We prove this by induction, showing that some finite cover of the action on Zn−j is the product of the homogeneous action on Zn and the Anosov flows on Yn,...,Yn−j. By construction, the base k Tn case of j = 0 holds. Let A denote the subgroup of R defined by j=1 ker αj, and ∆n−j denote the set of weights for the action on Zn−j. Therefore, assume that we have the desired structure on Zn−j+1, and note that Zn−j is a space with Yn−j as a factor and Zn−j+1 as a fiber by construction. One of the key differences between direct product and skew-product actions is that for direct products, the action on the factors has a canonical lift to a flow on the total space. The following lemma provides this.

j \ j Lemma 19.8. For the action of A×R on Zn−j, Lj := ker β ⊂ A×R is a one-dimensional β∈∆n−j+1 subspace.

Proof. We first show that it is at most one-dimensional. Indeed, if dim(L) > 1, then ker αn−j ∩ L would either be L itself or a co-dimension one subspace in L, and therefore nontrivial. But ker α∩L is the Starkov component, so it must be trivial. Therefore, dim(L) ≤ 1. k To see that dim(L) ≥ 1, choose a periodic orbit p of R and let z = πn−j+1(p). Choose any j −1 a ∈ (A × R ) \ ker αn−j such that a · p = p. Then a ∈ ZDiff1 (H y π (y)) and therefore some finite n power of a, a , coincides with the action of some h ∈ ker αn−j since by the induction hypothesis, Proposition 19.5 and Lemma 19.6, the action ker αn−j y Zn−j+1 is virtually self-centralizing. n −1 n −1 Therefore, a h fixes p and acts trivially on Zn−j+1. In particular, a h ∈ ker β for every n −1 β ∈ ∆n−j+1. Furthermore, a h 6= Id since a 6∈ H. Hence L is a nontrivial subspace.

The following lemma will follow from an adaptation of the standard Livsic theorem applied to the flow ψt (which lifts to the action of Lj). It is crucial that Lj satisfies Lemma 19.8, it will not hold for an arbitrary transverse subgroup. One cannot apply Livsic techniques directly for the L-action on X since it is only partially hyperbolic. Instead, we shall see that we can build a cocycle over ψt 106 which incorporates the derivatives at every point of the fiber, has trivial periodic data, but pay the price of it taking values in a Banach space of Hölder functions.

Lemma 19.9. There exists a Hölder continuous family of Hölder Riemannian metrics ||·||y on −1 ±α −1 −1 0 T (π (y)) such that every ` ∈ L and every α and −α holonomy fy,y0 : π (y) → π (y ) is an isometry. Proof. Fix a coarse Lyapunov distribution Eβ, ±α 6= β ∈ ∆. Consider the bundle over Y in which the fibers are given by Cθ(π−1(y)), the θ-Hölder continuous functions on π−1(y). Fix the unique ` ∈ L such that π((t`) · x) = ψt(π(x)) for every x ∈ X. Fix a Riemannian metric ||·|| on X. We may without loss of generality assume that ||·|| is invariant under the finite order elements of −1 ZDiff1 (H y π (y)) for every y ∈ Y , since the actions are all smoothly conjugate. Define a family θ −1 of sections Dβ(·, t), t ∈ R, so that Dβ(y, t) ∈ C (π (y)) in the following way:

−1 Dβ(y, t)(x) = log ||d(t`)|Eβ (x)|| for every x ∈ π (y).

Notice that Dβ satisﬁes the cocycle equation over the ψt-action:

Dβ(y, t + s)(x) = log ||d((t + s)`)|Eβ (x)||

= log ||d(t`)|Eβ (x)|| + log ||d(s`)|Eβ (t` · x)||

= Dβ(y, t)(x) + Dβ(ψt(y), s)((t`) · x). θ −1 We will show that there exists a section H(·) such that H(y) ∈ C (π (y)) and Dβ(y, t)(x) = H(ψt(y))(t` · x) − H(y)(x). We follow the standard Livsic argument. Suppose that p ∈ Y is a −1 −1 ψt-periodic orbit, so that ψT (p) = p for some T . Then T ` : π (p) → π (p) is a diffeomorphism commuting with the H-action on the fiber, and is in the kernel of every weight β 6= ±α. We may therefore without loss of generality assume that it is an isometry. In particular, we get that Dβ(p, T ) ≡ 0 whenever ψT (p) = p. The rest now follows from the standard Livsic construction applied to ψt on Y , with the function Dβ as a cocycle. That is, we choose a dense orbit and prove that the cocycle restricted to the orbit is uniformly Hölder continuous to get an extension. This implies that there exists H as described. β If π(x) = y, define a new norm on Ex :

−H(y)(x) ||v||β,x = e ||v|| . β Then notice that if v ∈ Ex , then

−H(ψt(y))(t`·x) ||d(t`)(v)||β,(t`·x) = e ||d(t`)(v)||

−H(ψt(y))(t`·x)+Dβ (y,t)(x) H(y)(x) = e ||v|| = e ||v|| = ||v||β,x . One may repeat this procedure for every β. Since H commutes with L, there is an L-invariant metric along the H-orbits. Since T (π−1(y)) is the direct sum of each Eβ together with the tangent bundle of the H-orbits, we may put the metrics together to build a metric on which L acts by isometries. Then the ±α-holonomies are automatically isometries as well (since they are limits of isometries).

We now show that the induction hypothesis holds on Zn−j, which concludes the proof of Theorems 1.10 and 1.11 by induction on j. 107 Proposition 19.10. There exists a connected finite cover Yñ−j of Yn−j with lifted flow ψ˜t and a k finite-to-one local diffeomorphism T : Yn−j × Zn−j+1 → Zn−j which intertwines the R actions on Yñ−j × Zn−j+1 and Zn−j. That is, if b ∈ Lj projects to ψ1 and a ∈ ker αn−j, then:

T (ψ˜t(y), a · x) = (tb + a) · T (y, x). u s Proof. We proceed as in the proof of Lemma 19.7. Notice that given a W ,W , ψt-path on Yn−j −1 starting at yn−j (where yn−j satisfies πn−j+1(yn−j) = Zn−j+1) one may similarly define holonomies α −α by lifting the path to a W n−j ,W n−j ,Lj-path on Zn−j. Rather than defining an action on the centralizer, because the ψt-orbit foliations now have a canonical lift, we may consider the holonomy −1 maps fρ : Zn−j → π (e(ρ)), which intertwine the action of ker αn−j on the fibers. r0 0 We claim that the maps fρ are C (where r = (1, θ) or ∞ if r = 2 or ∞, respectively). Indeed, recall the proof of Lemma 8.4, and notice that the roles of W α and Wdα may be reversed as long α k as Wd is the slow foliation of some a ∈ R acting on Zn−j. Choose any a ∈ Lj from Lemma 19.8 0 r and perturb it to a regular element a acting on Zn−j. This shows that the α-holonomies are C s u r k along Wa and Wa . They are C along R -orbits since the foliation is invariant. Therefore, by the standard application of Journé’s theorem, the holonomies are Cr0 . u s Finally, notice that if σ is a cycle in the W ,W , ψt-orbit foliations, then fσ : Zn−j → Zn−j centralizes the ker αn−j action. Therefore, since the action on Zn−j is virtually self-centralizing by the induction hypothesis, Proposition 19.5 and Lemma 19.6, there is a finite index subgroup of u s the group of W ,W , ψt-orbit cycles at yn−j for which the map fσ is trivial. As in the proof of u s Lemma 19.7, we quotient the space of all W ,W , ψt-orbit paths based at yn−j by this finite index subgroup, which is by construction a finite extension of Yn−j. Call this space Yñ−j. u s Finally, we define the map T . Given a W ,W , ψt-orbit path ρ based at yn−j and a point x ∈ X, let T¯(ρ, x) = fρ(x). Then by construction, T¯ descends to a map on Yñ−j and satisfies the desired properties. 20. Applications of the Main Theorems ` Proof of Corollary 1.8. Let Z y X be a transitive, totally Cartan action. Let X˜ denote the ` suspension space of X, equipped with a transitive, totally Cartan action of R . By Lemma 17.2, ` the Starkov component of R acts through a free torus action. In particular, it must be a rational ` subtorus in the torus projection π : X → T . Let k = ` − dim(S), and choose any rational k- ` dimensional subspace V transverse to S. Then let a1, . . . , ak ∈ Z be generators of this subspace. Notice that since the first return action of S is by a finite group, and we wish to embed only a finite index subgroup, it suffices to consider the case of trivial Starkov component. ` Rather than suspending Z y X, we suspend the action of ha1, . . . , aki. Call the resulting space k Y˜ , and notice that by construction, the R action on Y˜ has trivial Starkov component. One may m suspend further if necessary to get a self-centralizing R action, m ≥ k, on a space Y¯ in which the k action of R embeds as a rational hyperplane (as in the start of the proof of Theorem 1.11). The space Y¯ is still a suspension of an action on X (since it is the suspension of a discrete action, on m an existing suspension space) with a factor map π¯ : Y¯ → T . Then, after passing to a finite cover if necessary, Y¯ is isomorphic to the product of finitely many Anosov flows on 3-manifolds and a homogeneous action. We claim that the generators of each Anosov flow on a 3-manifold, and the span of the generators m ¯ of the homogeneous action, are all rational subspaces in R . Indeed, notice that Y = Y1 ×· · ·×Yn × G/Γ. Then Yi carries a generator of the Anosov flow which must be closed in the torus factor of Y¯ since Yi is an Anosov flow (its closure must be a subtorus, and if its dimension was more than one, 108 the rank of the factor would be greater than one). Therefore, the generator of the flow on the space Yi is rational. Similarly, the generators of the action of G/Γ must correspond to a rational subspace m of the torus factor T , as it must be exactly complementary to the sum of the directions for each Yi. This immediately implies that every Yi, as well as the homogeneous flow on G/Γ, are suspensions. m After taking a finite index subgroup of Z we may without loss of generality assume that the smallest rational generators of the Anosov flows and action on G/Γ are the standard generators of m Z . Thus the first return action must be isomorphic to a product of Anosov diffeomorphisms of 2-manifolds, and an affine Anosov action on a homogeneous space. Since the reductive component of the homogeneous space must be trivial by the Anosov condition, and any Anosov diffeomorphism of an orientable 2-manifold must be on a torus, we conclude the corollary.

Proof of Corollary 1.12. First, every homogeneous action preserves a volume, and the Ledrappier- Young formula implies that the volume maximizes entropy (since it maximizes Hausdorff dimension and the Lyapunov exponents are constant). So we must show that if a volume maximizes entropy, then the action is homogeneous. Notice that if an action is homogeneous, then the centralizer is affine by Lemma 19.4 (it is not difficult to prove that once a centralizing element is C∞, it is affine). Therefore, we may assume without loss of generality that the action is self-centralizing. We may therefore apply Theorem 1.10, so that the action is a product of a homogeneous action and Anosov flows on 3-manifolds. Let π : X → Y be one of the rank one factors of the action, equipped with a corresponding Anosov flow ψt. The measure of maximal entropy for any transitive Anosov element is unique by [10]. Since a is Anosov, the induced action of a on Y cannot be trivial, so a = ψt for some t 6= 0. Without loss of generality, we assume t = 1. Notice that since the topological entropy of a is the sum of the topological entropy of a on each factor, and each factor has a measure of maximal entropy, µ must be the product of those measures. Since the element a is transitive on X, ψ1 is transitive on Y . So by [10, Theorem 5.4], the measure induced by µ on Y is the measure of maximal entropy. Since it is a C∞ factor, it is a volume. Therefore, by [70], the flow on Y is homogeneous.

Proof of Corollary 1.13. Suppose G is a semisimple real linear Lie group of noncompact type, Γ ⊂ G be a lattice such that Γ projects densely onto any PSL(2, R) factor, and M = G/Γ be the corresponding G-homogeneous space. Passing to a subgroup of finite index, we may assume that Γ is torsion-free. If K denotes a maximal compact subgroup of G, then Γ acts freely and properly discontinuously on K \ G, the symmetric space attached to G. The latter is aspherical, and thus Γ = π1(K \ G/Γ). By the Serre long exact sequence for a fibration, there is a surjection whose kernel is π1(K), α : π1(G/Γ) 7→ π1(K \ G/Γ) = Γ. 2 k Consider a C transitive, totally Cartan R -action on M. After passing to a finite cover, by Theorem 1.11, M can be written as a product of 3-manifolds M1,...,Mk and a homogeneous space H/Λ. Hence π1(G/Γ) = π1(M1) × ... × π1(Mk) × π1(H/Λ). Set Λi = α(π1(Mi)) and Λk+1 = α(π1(H/Λ)). Then each Λi commutes with every Λj for i 6= j. Suppose that 1 6= λ ∈ Λi ∩Λj for i 6= j. Then λ commutes with all the Λk, and hence is in the center of Γ. As Γ is Zariski dense in G by Borel’s Zariski density theorem, the Zariski closure (over R) of hλi is in the center of G which by assumption of linearity of G is finite. As Γ is torsion free, λ = 1. We conclude that Γ = Λ1 × ... × Λk+1. Next let Gi be the Zariski closure of Λi in G. Then all Gi commute with all Gj for i 6= j, and again Gi ∩Gj = {1} for all i 6= j, by the same argument as for the Λi. Hence G = G1 ×...×Gk ×H is a splitting such that for all i, Γi ⊂ Gi. Since Γ is a lattice in G, each Λi is a lattice in Gi for all i. 109 Moreover Λi is the fundamental group of the compact manifold Ki \ Gi/Λi for all i where Ki ⊂ Gi is a maximal compact subgroup Gi. Then Λi = π1(Ki \ Gi/Λi) as above. We recall that closed manifolds supporting codimension 1 Anosov flows are aspherical [2, Lemma 5.5, p.147]. Hence the cohomological dimension of the π1(Mi) is 3 for all i ≤ k [8, VIII.8 Proposition 8.1]. Moreover, as the π1(Mi) and the Λi are Poincare duality groups (as fundamental groups of closed manifolds), the cohomological dimension of the Λi is at most that of the π1(Mi) for all i ≤ k [3, Theorem 3.5]. Since the Λi are fundamental groups of the closed aspherical manifolds Ki \Gi/Λi for all i ≤ k, the Ki \ Gi are symmetric spaces of dimension at most 3 with Gi a semisimple group 2 3 of the noncompact type. There are only two possibilities, the real hyperbolic spaces H or H . The first case is explicitly excluded since we do not allow Γ to project discretely into any PSL(2, R). In the second case, Mi would be covered by SO(3, 1) which contradicts that the Mi have dimension 3. In conclusion, we see that G/Γ does not have any rank 1 factors, and G/Γ is homeomorphic to H/Λ. By Mostow rigidity [53, Theorem A’], G is isomorphic to H and Γ to Λ. Applying the k 1,θ resulting conjugacy, our R totally Cartan action is C -conjugate to a homogeneous subaction of k k G on G/Γ by a subgroup A of G (up to an automorphism of R ). Since the R -action is Cartan, A is a split Cartan of G. Since the coarse Lyapunov spaces are 1-dimensional, so are the root spaces of G. Hence G itself is R-split. ∞ Finally, if the action is C , so is the conjugacy. Proof of Corollary 1.14. Let G be a semisimple Lie group, and G y X be a locally free action of G such that the restriction of the action to a split Cartan subgroup A ⊂ G is totally Cartan. Notice then that the bundle which is tangent to the G-orbits as an invariant sub-bundle yields an A-invariant subbundle of TX. Since the action is by a Cartan subgroup of G, the root splitting of G with respect to A must coincide with the coarse Lyapunov splitting, and the centralizer of the Cartan must be trivial. Furthermore, G has no compact factors. Therefore, G is R-split, and the root subgroups of G are coarse Lyapunov subgroups. Since the roots of a semisimple Lie algebra span the dual space to any Cartan subalgebra, the Starkov component must be trivial. Therefore, some finite cover of the action is C1,θ-embedded in a direct product of rank one systems and a homogeneous action. We claim that the Anosov flow t k ϕi : Yi → Yi is a homogeneous flow as well. Indeed, let a ∈ R be a the generator of such a flow, and notice that since the roots of G generate the dual space, there exists a root α such that α(a) 6= 0. ±α Then the pair ±α is unique, since the flow on Yi splits as a direct product. Since the flows η must be the action of unipotent subgroups of G, and the factor Yi is generated by the homogeneous α −α flows ta, ηt and ηt , they must generate a copy of (a cover of) PSL(2, R). So the factors Yi are homogeneous. Finally, notice that the relations on the group G automatically determine the commutator and symplectic relations in the path group Pˆ used in the proofs of Part II (ie, they determine the pairwise cycle structures (Definition 14.1 for weights corresponding to roots). Therefore, the group G embeds into the group giving the homogeneous structure provided by the dynamics, and the G-action, as well as the A-action, is homogeneous. Proof of Corollary 1.15. By Corollary 1.8, the manifold X must be a nilmanifold. Then by [6, Corollary 1.8(2,3)], a finite index subgroup of Γ is smoothly conjugated to an affine action.

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