Axiom a Diffeomorphisms Derived from Anosov Flows

JOURNAL OF MODERN DYNAMICS doi: 10.3934/jmd.2010.4.1 VOLUME 4, NO. 1, 2010, 1–63

AXIOM A DIFFEOMORPHISMS DERIVED FROM ANOSOV FLOWS

CHRISTIAN BONATTI AND NANCY GUELMAN (Communicated by Amie Wilkinson)

ABSTRACT. Let M be a closed 3-manifold, and let Xt be a transitive Anosov flow. We construct a diffeomorphism of the form f (p) Y (p), where Y is = t(p) an Anosov flow equivalent to X . The diffeomorphism f is structurally stable (satisfies Axiom A and the strong transversality condition); the non-wandering set of f is the union of a transitive attractor and a transitive repeller; and f is also partially hyperbolic (the direction R.Y is the central bundle).

1. INTRODUCTION 1.1. Diffeomorphisms versus vector fields. Classic dynamical systems are those associated with differential equations (i.e., vector fields) and those defined by diffeomorphisms. Diffeomorphisms and vector fields are such deeply related theories that a result in one theory often has a counterpart in the other. For example, vector fields in dimension n share properties with diffeomorphisms in dimension n or n 1: − Via the flow. The time-one map of the flow of X is a diffeomorphism of the • same regularity as the vector field of the same manifold. However, most diffeomorphisms are not the time-one map of a flow; (for example, see [24]). Via the transverse structure. Poincaré used the first-return map of the flow • to a cross-section Σ: the first-return map is a local diffeomorphism defined on Σ. Note that the dimension of Σ is dimM 1, where M is the ambient − manifold. Conversely, every diffeomorphism f : N N may be viewed as → the first-return map of a flow to a global cross-section; this flow is called the suspension of f . This double similarity also emerges when defining the notion of hyperbolicity. A compact invariant set K for a diffeomorphism f is hyperbolic if the tangent space of the manifold splits into the sum of two, D f -invariant sub-bundles, TM E s E u, where the vectors in the stable bundle, E s, are uniformly con- |K = ⊕ tracted by D f whereas those in the unstable bundle, E u, are uniformly expanded.

Received November 15, 2008, revised January 15, 2010. 2000 Mathematics Subject Classiﬁcation: Primary: 37D20; Secondary: 37D30. Key words and phrases: Anosov ﬂows, Axiom A diffeomorphism, partial hyperbolicity, Birkhoff sections, perturbations. Partially supported by Université de Bourgogne, the IMB, the grant EGIDE de la Région Bour- gogne, PEDECIBA and Universidad de la República, the DySET project of MathAmSud and the DynNonHyp project of AMR.

If we now consider a vector field X , the same definition of hyperbolicity holds— albeit only in the specific case when K is a singular point. If K is a nonsingular compact set invariant under the flow of X , the direction of the flow is neither expanded nor contracted, so one needs to assign a specific role to the direction of the flow: K is hyperbolic if TM E s RX E u, where E s and E u are invariant |K = ⊕ ⊕ under the flow and uniformly contracted and expanded, respectively. Nevertheless, both notions of hyperbolicity (for flows or diffeomorphisms) are related through the transverse structure:

If Λ is hyperbolic for X , the ﬁrst-return map of X to a cross-section Σ is • hyperbolic at Λ Σ; ∩ Conversely, the suspension of a hyperbolic set for a diffeomorphism is a • hyperbolic set for the ﬂow.

However, if K is a nonsingular hyperbolic set for a vector ﬁeld X , then the diffeomorphisms f X ( ,t), t R cannot be hyperbolic. t = · ∈

1.2. Anosov diffeomorphisms and vector fields. A vector field X (a diffeomorphism f ) on a compact manifold M is called Anosov if the whole manifold M is a hyperbolic set for X (resp. f ). Thus, the suspension of an Anosov diffeomorphism is an Anosov flow. This is a very rigid structure: every vector field Y in a C 1-neighborhood of X is an Anosov vector field topologically equivalent to X : there is a homeomorphism of M mapping the orbits of Y to the orbits of X . The same holds for Anosov diffeomorphisms. One may say that the dynamics of Anosov vector fields and diffeomorphisms are well understood, yet many questions remain open. Whereas the two cases (diffeomorphisms and vector fields) are highly related, answers to the following questions may nevertheless be case dependent:

What manifolds admit Anosov diffeomorphisms? It is now well-known that • codimension-one (i.e., a codimension-one unstable bundle) Anosov diffeomorphisms are conjugate to a linear Anosov diffeomorphism on a torus T n. (See [10] and [23]) What manifolds admit Anosov vector fields? For codimension-one Anosov • flows (i.e., the stable bundle has dimension one) on closed manifolds M with dimM 4, Verjovsky conjectures that they are topologically equiva- ≥ lent to the suspension of an Anosov diffeomorphism (and hence of a linear toral automorphism) (see [32, 13, 30] for partial answers to this conjecture). Is any Anosov diffeomorphism transitive? For flows the answer is negative: • Franks and Williams ([12]) constructed a non-transitive Anosov flow on 3- manifolds. There are many examples of transitive Anosov vector fields on 3-manifolds, • such as the classic example of geodesic flow on a compact surface with negative curvature and suspension of T 2 hyperbolic linear automorphisms.

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Yet, there are exotic examples [5], so classiﬁcation remains an open problem (see [1, 9] for works in this direction). The corresponding case for diffeomorphisms is Anosov diffeomorphisms in dimension 2, which are com- pletely understood as conjugate to linear torus automorphisms associated with an element of SL(2,Z).

1.3. Stability, hyperbolicity and classification. Hyperbolicity is key to structural stability: If K is a hyperbolic compact invariant set for a vector field, X , it ad- 1 mits a (hyperbolic) continuation KY for every vector field Y C -close to X ; KY is a hyperbolic invariant compact set for Y that is homeomorphic to K through homeomorphism hY . This map is a topological equivalence; that is, hY maps the oriented orbits of Y in KY to oriented orbits of X on K ). Similarly, hyperbolic sets of diffeomorphisms admit hyperbolic continuation. Moreover, according to [27, 28, 22, 19], a diffeomorphism f (or a vector field X ) is structurally stable if and only if it satisfies Axiom A and the strong transversality condition, which may be expressed as follows: The chain-recurrent set is hyperbolic, and the stable and unstable manifolds of every pair of chain-recurrent points are transverse. Succinctly, diffeomorphisms or vector fields satisfying Axiom A and the strong transversality condition are said to be hyperbolic. Hyperbolic dynamical systems admit a satisfactory topological and ergodic description. For this reason one could foresee a complete classification of hyperbolic dynamical systems. Such a classification has been achieved in dimension two for vector fields [26] and for diffeomorphisms [5] but remains an open problem in the case of higher dimensions. Global classification seems beyond reach in the near term; hence, researchers tend to focus on specific families and to make the following assumptions: Simple dynamics. Consider Morse–Smale flows or diffeomorphisms in di- • mension 3; Rigid structures. Assume the existence of a non-periodic hyperbolic attrac- • tor. In that case, the basin of the attractor is an open region foliated by the stable manifolds of the points in the attractor: the strong transversality condition is therefore very restrictive (see [14] and [5, Théorème 2.3.4]). In dimension 3, [15] gives a complete classification (up to topological ≥ conjugacy) of structurally stable diffeomorphisms with a codimension-one attractor (i.e., the unstable manifold of any point of the attractor has codimension one).

1.4. Normal stability. However, the lack of hyperbolicity is just along the X - orbits: according to the terminology of [20], K is a normally hyperbolic C 1-lamination whose leaves are the X -orbits; then [20] shows that the lack of structural stability is “just along the X-orbits”: For every diffeomorphism g C 1-close to f , there is an invariant compact set, Kg that is a normally hyperbolic lamination 1 homeomorphic to K through a homeomorphism hg such that hg ghg− is a per- 1 turbation of f along the orbits of X ; for every x K , the point h gh− (x) is a ∈ g g JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 4 CHRISTIAN BONATTI AND NANCY GUELMAN point in the X -orbit of x close to f (x). More precisely, there is a continuous map t : M R close to 1, such that: → 1 h gh− (x) X (t(x),x). g g = 1.5. Perturbations of the time-one map of an Anosov flow. Let X be a transitive Anosov vector field on a compact manifold, and let f be the time-one map of its flow. As explained in Section 1.4, f is neither hyperbolic nor structurally stable; however, according to [20], any C 1-perturbation of f is topologically conjugate to a perturbation of f along the orbit of X . Evidently, f is not “so far from being hyperbolic” nor “so far from being structurally stable.” This observation precipi- tated many questions on the dynamics of the perturbations of f , some of which are now understood while others remain open. For example, the time-one map of a geodesic flow of a constant negative curvature surface stands as the first instance of a non-hyperbolic, stably-ergodic diffeomorphism (see [17]), and this example has generated a large theory of stable ergodicity. A topological equivalent to stable ergodicity is robust transitivity, which states that every g C 1-close to f is transitive. The time-one map f of the flow of a transitive Anosov vector field X is not necessarily robustly transitive: If X is the suspension of an Anosov diffeomorphism, then f can be perturbed in order to break transitivity. However, there are no other known examples of transitive Anosov flows for which one can determine whether or not the time-one map of its flow is robustly transitive:

QUESTION 1.1. Let X be a transitive Anosov vector ﬁeld that is not conjugate to a suspension, and let f be the time-one map of its ﬂow. Is f robustly transitive? (The answer may depend on X ).

A positive answer to this question would provide an initial example of a robustly transitive diffeomorphism without periodic points. As a partial answer, it has been shown [2] that for any transitive Anosov vector field X , f admits C ∞- small perturbations that are robustly transitive. On the other hand, if f is not robustly transitive, some perturbation of f can break the limit set of f into several pieces. Each of these pieces is not a priori hyperbolic, but the lack of hyperbolicity is just in the direction of the vector field X ; hence, one could speculate that the direction of the flow could be modified in order to be expanded on some pieces and contracted on others. This intuitive question was formalized in 1975 by Palis and Pugh [25], who posed the following:

QUESTION 1.2. May the time-one map f : p X (1,p) of a transitive Anosov vec- 7→ tor ﬁeld X be C 1 approximated by Axiom A diffeomorphisms?

As was observed, the answer is positive when X is the suspension of a transitive Anosov diffeomorphism. The fact that f may be approximated by an Axiom A diffeomorphism seems to imply strong geometric restrictions, particularly on the attractors and repellers, and we provided partial answers to Question 1.2 in earlier works [16, 4].

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1.6. Results. In this paper we consider (not necessarily C 1-small) perturbations of the time-one map of the flow of X “in the direction of the flow;” that is, we consider diffeomorphisms of M that map each point of M to a point of its X - orbit. More precisely, we denote by E˜(X ) the set of diffeomorphisms f : M M → defined as follows: There is a C 1 map t : M (0, ) such that f (x) X (x,t(x)), x M. In • → +∞ = ∀ ∈ particular, f preserves every leaf of the 1-foliation generated by X ; The diffeomorphism f is partially hyperbolic and its central bundle is gen- • erated by X . We then define E (X ) as the set of diffeomorphisms f such that there is an Anosov flow Y topologically equivalent to X with f E˜(Y ). In [4], we proved ∈ that any Axiom A diffeomorphism in E (X ) contains at most one transitive attractor (and at most one transitive repeller) if X is not the suspension of a transitive Anosov diffeomorphism. In this paper we exhibit examples of Axiom A diffeomorphisms in E (X ) with only one transitive attractor and only one transitive repeller. Specifically, we prove the following:

THEOREM 1.3. Let X be a transitive Anosov ﬂow of a compact 3-manifold M. Then there exists a diffeomorphism f E (X ) such that: ∈ f satisﬁes Axiom A and the strong transversality condition (i.e., f is struc- • turally stable); Ω(f ) is the union of just a transitive attractor and a transitive repeller. • 1.7. Idea of the construction.

1.7.1. The case of a suspension. Let us first consider an Anosov vector field X on a compact manifold MA that is the suspension of an Anosov diffeomorphism A : N N. The manifold M is a fibration1 over S1, with fibers diffeomorphic → A to N. Each fiber is a global cross-section of X , and the first-return map on the fiber is A. The time-one map f X ( ,1) of the flow of X preserves each fiber = · and induces on each fiber a diffeomorphism smoothly conjugate to A. Selecting 2n fibers N , i 1,...,2n, (cyclically ordered) and perturbing g of f along the i = orbits of X preserves precisely the fibers Ni , effectively multiplying fibers N2i by a transverse contraction and fibers N2i 1 by a transverse expansion. Each point + outside of Ni is between an attracting fiber and a repelling fiber and the dynamic of g pushes the orbits from the repeller to the attractor.

1.7.2. The general case. A flow is topologically equivalent to a suspension if and only if it admits a closed embedded global cross-section. Fried [11] noticed that any transitive Anosov flow on 3-manifolds “almost admits global cross-sections”. This observation is based on the concept of Birkhoff section, which is an embedded surface with boundary B , M such that: → the interior B ∂B of B is transverse to the vector field X ; • à the boundary ∂B is the union of finitely many periodic orbits of X ; • 1called the mapping torus by topologists

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there is T 0 such that for every point x M there is t (0,T ] with X (x,t) • > ∈ ∈ ∈ B. Fried [11] built (infinitely many) Birkhoff sections2 for any transitive Anosov flow on a 3-manifold M. He also proved that the first-return map, P, defined on the interior of B induces a pseudo-Anosov diffeomorphism P˜ on the closed surface, B˜, where B˜ is obtained from B by replacing each boundary component with a point. Under that interpretation, Fried noticed that X looks like the suspension of a pseudo-Anosov homeomorphism, and he described a simple surgery on the suspension of P˜ for reconstructing the flow X . It would therefore be tempting indeed to try to construct an Axiom A diffeomorphism close to the time-one map of the flow of X in the same fashion as was done in the case of a suspension flow, and, in fact, that is just what we propose in this paper. Here we consider a transitive Anosov vector field on a compact 3-manifold M endowed with an embedded Birkhoff section B0. By pushing a little bit on B0 along the flow of X , one obtains another Birkhoff section B1 “parallel to B0,” whose interior is disjoint from B but shares the same boundary, ∂B ∂B . 0 1 = 0 We repeat this construction for suspension out the boundary of B0 and B1, which leads to a diffeomorphism, f , defined on M ∂B such that B is an at- ext à 0 0 tractor and B1 is a repeller. However, the diffeomorphism cannot be extended to ∂B0 because both Birkhoff sections intersect along the boundary. We then replace fext in a tubular neighborhood of each component of ∂B0 with a local model, f , which is a diffeomorphism of R2 S1. It is the construction of this mod × local model that represents the primary challenge and forms the technical sub- stance reported in this paper. Ultimately, we smoothly glue fext with fmod. 1.8. Dynamics of our perturbation. Starting with a transitive Anosov flow X and a Birkhoff section B , we build a structurally stable diffeomorphism f E (X ) 0 ∈ whose non-wandering set Ω(f ) consists of an attractor A and a repeller R. Let N denote the closed surface obtained from B0 by replacing each boundary component of B0 with a point.

According to Fried [11], the first-return map PB0 of the flow of X on the interior of B induces on N a pseudo-Anosov homeomorphism P : N N whose 0 N → singular points correspond precisely to the boundary components of B0. By a classical construction consisting in opening the unstable separatrices of the singular points, one builds a structurally stable diffeomorphism of N (called derived from the pseudo Anosov map PN ), having a (non-periodic) hyperbolic attractor. The attractor A of f is conjugate to this attractor. In the same way, the repeller R of f is conjugate to the one obtained by opening the stable separatrices of the singular points of PN . However, the attractor A and the repeller R are not contained in a surface transverse to the vector field X . Rather than obtaining the attractor A by means of the classical construction derived from the pseudo Anosov-map, the attractor A is instead obtained from

2The sections built in [11] may not be embedded: their interior is embedded but their boundary may be just immersed. However, as noticed in [7], one can modify slightly the construction in [11] in order to get embedded Birkhoff sections.

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PB0 by opening the unstable separatrices in the direction of the flow of X (which is transverse to the surface Int(B0)); the repeller R is obtained by opening the stable separatrices in the direction of the flow of X (see Section 9). The attractor A and the repeller R are almost transverse sections of the flow: almost all orbits of X cut successively A and R in a point. The only orbits with a different behavior are those in the unstable and stable manifolds of the boundary components of B0: those in the unstable manifold are cutting A twice before cutting R, and those in the stable manifold are cutting R twice before cutting A. One recovers B0 from A by collapsing every segment of the X -orbit disjoint from R and with both endpoints on A.

1.9. Progress toward a positive answer to the question of Palis and Pugh. The diffeomorphism f we built depends strongly on the choice of the Birkhoff section B0. Each choice of B0 (up to moving B0 along the orbits of X ) leads to a different diffeomorphism f (up to topological conjugacy by homeomorphisms isotopic to the identity). Given a Birkhoff section, our construction does not lead to a C 1-small perturbation of the time-one map of the ﬂow; for this reason, at different steps of the construction, we did not mind choosing large perturbations even if those steps could be done by C 1-small perturbations. However, we do not know whether or not, given a size of perturbation, one can choose a Birkhoff section such that our procedure would lead to a C 1-small perturbation—and hence answer the Palis–Pugh question.

QUESTION 1.4. Is it possible to find a transitive Anosov flow that admits a sequence of Birkhoff sections B0,n in such a way that the corresponding maps, fn, converge to the time-one map of the flow in the C 1-topology?

We now provide an encouraging result in this direction. In a non-published work, Shaobo Gan studied perturbations of the diffeomorphism F (A,ρ) of the torus T 3 T 2 S1 obtained as the product of an Anosov = = × diffeomorphism A of T 2 by a rotation ρ. In an analogous result to ours [4], Gan shows that if f is a perturbation of F that is structurally stable, and if f has more than one attractor, then the attractors and the repellers of f are (topological) torus T 2, (topologically) transverse to the S1 fibers, and f , restricted to the torus, is conjugate to an Anosov map. However, our construction leads to perturbations of F with exactly one hyperbolic attractor and one hyperbolic repeller; the diffeomorphisms are derived from Pseudo-Anosov diffeomorphisms restricted to the attractor and to the repeller. This construction depends on the choice of a Birkhoff section B0 of the 1 1 S -fibration that is a compact surface B0 whose interior is transverse to the S - fibers and whose boundary consists of finitely many fibers: then, exactly as in our construction, it consists of pushing F along the fibers. In this similar problem, which is geometrically much more simple, we think (together with S. Gan) that it is possible to choose a sequence of Birkhoff sections B0,n in such a way that the corresponding maps fn converge to F in the C 1-topology. The idea is that it is possible to fix the boundary of the Birkhoff

JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 8 CHRISTIAN BONATTI AND NANCY GUELMAN sections and then choose the Birkhoff sections to have arbitrarily large linking numbers. The same holds for Anosov flows that are suspensions of Anosov torus automorphisms. This allows us to formulate: 2 CONJECTURE 1.5. Let A be an Anosov diffeomorphism of the torus T , let X A be the suspension of A in a compact manifold MA, and denote by f A the time-one mapoftheflowof X A. Then there are structurally stable diffeomorphisms fn con- 1 verging to f A in the C -topology, such that the non-wandering set of fn consists of exactly one attractor An and one repeller Rn, but An and Rn are not homeomorphic to a torus. In general, we think that the C 1-distance of our construction with the time- one map of the flow depends mostly on the the ratio ρ(B) associated with the Birkhoff section B by π(γ) ρ(B) sup , = γ ∂B ℓ(B,γ) ∈ where π(γ) is the period of a periodic orbit in the boundary of B, and ℓ(B,γ) is the linking number of B at γ (see Section 5).

QUESTION 1.6. Does an Anosov vector field X exist that is not topologically equivalent to a suspension and a sequence, Bn, of Birkhoff sections of X such that ρ(Bn) tends to 0? 1.10. Progress towards a negative answer to the question of Palis and Pugh. Our result in [4] was an attempt to give a negative answer to Question 1.2 by try- ing to find a contradiction in the rigid geometrical properties satisfied by the attractor and the repeller of an Axiom A perturbation of X ( ,1). These geometrical · properties led to the construction provided in this paper. We think it is possible that our examples are essentially the only ones possible; namely, that every Axiom A diffeomorphism in E (X ) should be strongly related to one of our examples, having a unique attractor obtained from our example by employing some simple “derived from” procedure; likewise for the repeller; and with the basic pieces in between attractor and repeller ultimately removable. In particular, we think that if there is an Axiom A, g E (X ), C 1-close to the ∈ time-one map X ( ,1), then there is another C 1-small perturbation of X ( ,1) that · · is conjugate to one of our examples. Let fn be such a sequence of our examples converging to X ( ,1) in the C 1-topology, and let B be the associated Birkhoff · 0,n sections. We think that, in that case, the ratio ρ(B0,n) should tend to 0, in which case we then have some feeling or intuition that the sections B0,n should converge to a foliation that is tangent to the bundle E s E u. However this bundle ⊕ is integrable only in the case when X is a suspension (in particular, we surmise that the answer to Question 1.6 is negative).

1.11. Our examples and the hyperbolic dynamics classiﬁcation problem. Our result provides a large new class of examples of hyperbolic (i.e., structurally stable) diffeomorphisms. As explained in Section 1.3, strong transversality is a very restrictive condition when the diffeomorphism admits a nontrivial hyperbolic

JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 AXIOM A DIFFEOMORPHISMS DERIVED FROM ANOSOV FLOWS 9 attractor or repeller. In dimension 3 there are few examples of structurally stable diffeomorphisms having nontrivial codimension-2 attractors; in fact, all the examples we know of are obtained (up to topological conjugacy) by completing the dynamics of structurally stable diffeomorphisms of a normally hyperbolic surface3. Our construction provides an infinite class of new examples: to every transitive Anosov flow X on a 3-manifold, our construction associates infinitely many different conjugacy classes of structurally stable diffeomorphisms with two additional and important properties: They are partially hyperbolic; and • Their non-wandering sets are the union of just one attractor and one re- • peller. These two properties define two classes of structurally stable diffeomorphisms for which it seems reasonable to achieve a complete classification. PROBLEM 1.7. Give a classification (up to topological conjugacy) of structurally stable diffeomorphisms on closed 3-manifolds whose non-wandering set consists of one attracting basic set and one repelling basic set. (Our examples show that this problem is far from trivial). PROBLEM 1.8. Let f : M 3 M 3 be a structurally stable diffeomorphism with a → partially hyperbolic splitting, TM E s E c E u, = ⊕ ⊕ where dimE s dimE c dimE u 1. = = = Is it true that f satisfies one of the following possibilities? f is an Anosov diffeomorphism; • M T 3 (up to a finite covering) and E c is tangent to an invariant circle • = foliation; E c is tangent to a 1-foliation F c generated (up to topological equivalence) • by an Anosov vector field X , verifying that each leaf is invariant under an iterate f n, n 0. > (See [6] and [18] for results on classification of partially hyperbolic diffeomorphisms on 3-manifolds).

2. DETAILED OUTLINE OF THE PROOF 2.1. Normal form for an Anosov vector field and a Birkhoff section. Let X be a transitive Anosov vector field on a closed 3-manifold M endowed with an embedded Birkhoff section B0. The first step of our construction consists of modify- ing the pair (X ,B0) in such a way that it coincides with an explicit model in local

3In fact, Grines asked to the ﬁrst author if there are structurally stable diffeomorphisms on 3-manifolds having a hyperbolic attractor not conjugate to a hyperbolic attractor on a compact surface. Unluckily, the attractors in our examples are conjugate to surface attractors, so do not answer to Grines’ question.

JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 10 CHRISTIAN BONATTI AND NANCY GUELMAN coordinates, namely, in tubular neighborhoods of every component of ∂B0. We now offer a glimpse into the difficulties we had to overcome: As explained in the introduction, by pushing B along the flow we build • 0 another Birkhoff section B such that B B ∂B ∂B . That is not so 1 0 ∩ 1 = 0 = 1 easy because the definition of Birkhoff section in [11] does not describe the local behavior of B0 in a neighborhood of a component γ of ∂B0. In particular, the definition does not ensure that the return time of the orbits of X on the interior of B0 is bounded away from 0. The pair of parallel sections allows us to build a diffeomorphism f on M • ext à ∂B that is a perturbation of X ( ,1) along the flow. In order to complete the 0 · construction, we then needed to glue fext with a local model, fmod, defined on R2 S1 by a barycentric isotopy along the orbits of the flow, in a tubular × neighborhood D2 S1 of each component γ of ∂B ; this requires that X and × 0 B0 coincide in a tubular neighborhood of γ with a model vector field and a model Birkhoff section. As explained in the item above, we needed to modify slightly the vector • field X in neighborhoods of the periodic orbits, which are components of ∂B0: Why does the new vector field have a Birkhoff section close to B0? Section 4 of this paper develops a theory of Birkhoff sections for solving these challenges by initially restricting our study to sections B0 satisfying an extra condition that we call tame Birkhoff section. One may verify that the proof in [11] leads to our notion of tame Birkhoff section. We consider (tame) Birkhoff sections up to isotopies along the orbits of the vector field. We define the linking number of a Birkhoff section at a boundary component. We verify that the linking number is the only invariant, up to isotopy along the orbits, for local topological tame Birkhoff sections in a neighborhood of a boundary component (Lemma 4.12). Lemma 4.13 shows that given a Birkhoff section, B0, a component γ of ∂B0, and a local Birkhoff section, S, defined in a neighborhood of γ having the same linking number as B0, B0 can then be modified in a neighborhood of γ by an isotopy along the orbit of X such that the new section coincides with S in a (smaller) neighborhood of γ. As a consequence, Lemma 4.16 shows that every (tame) topological Birkhoff section is flow isotopic to a smooth Birkhoff section. Thus, if X and Y are two Anosov vector fields topologically equivalent through a homeomorphism h, each tame Birkhoff section B0 of X corresponds to a Birkhoff section of Y that is flow equivalent to h(B0). Section 5 exhibits a local model (Xmod,Bmod) of a local Birkhoff section of linking number equal to 1: X is a vector field on R2 S1 with affine coordinates, mod × and Bmod is a regular half-helicoid. This model induces models for every linking number by passing to a finite covering (for obtaining any positive linking numbers) and considering symmetries (for obtaining negative linking numbers). We conclude Section 5 by stating (Corollary 5.7): Given any (tame) Birkhoff section S of a given transitive Anosov vector field X , one may build a vector field

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Y topologically equivalent to X through a homeomorphism h and a pair of parallel Birkhoff section B and B (i.e., B B ∂B ∂B ) such that B and B 0 1 0 ∩ 1 = 0 = 1 0 1 are flow equivalent to h(S) and such that the triple (Y ,B0,B1) is in normal form; that is, (Y ,B0) coincides (up to finite covering) to the models Xmod,Bmod in local coordinates in a neighborhood of any component of ∂B0, and B1 is parallel to B0 (see Definition 5.6)). Up to changing the vector field X by a perturbation Y , we henceforth assume that (X ,B0,B1) is in normal form.

2.2. The construction far from the boundary of the Birkhoff section. Consider a transitive Anosov vector field endowed with a pair of parallel Birkhoff sections B and B that are in normal form at each component of ∂B ∂B . We fix some 0 1 0 = 1 δ 0, and we consider the union of the X -orbits segments of length 2δ centered > at the points of the interior B0 and B1. This defines two regions, Vext and Uext of M ∂B . As B and B are in normal form, one may choose δ 0 such that these à 0 0 1 > regions are disjoint. Section 6.1 builds a diffeomorphism f on M ∂B , preserving each X or- ext à 0 bit such that B0 and B1 are invariant under fext, and fext coincides on Bi with 1 the first-return map of the X -orbits on Bi . Furthermore, fext and fext− are affine 1 contractions by a factor α− 1 on the X -orbits segments in U and V , re- < ext ext spectively; hence, Uext is an attracting region and Vext is a repelling region. One chooses α close to 1 so that the contraction or expansion along X is smaller than the contraction or expansion due to the hyperbolicity of X , which corresponds to a lower bound of the return times on B0 and B1; this allows us to obtain a partially hyperbolic final map. Moreover, we require that the image under fext of the complement of Vext be contained in Uext: this is possible because one does not require that fext be close to the time-one map of the flow of X . This last hypothesis will help ensure that the limit set of the orbits of our final map f will be contained in U V , out- ext ∪ ext side of an arbitrarily small neighborhood of ∂B0.

2.3. Reduction to the construction of a local model. We consider R2 S1 en- × dowed with the model vector ﬁeld X log2 x ∂ y ∂ ∂ and with model mod = ∂x − ∂y + ∂z Birkhoff sections S0 and S1 that are the two standard³ half´ helicoids: the union S S is the standard helicoid, i.e., the union of straight lines in R2 {z} rotating 0 ∪ 1 × with constant speed when z runs along S1. Recall that (X ,B0,B1) is in normal form and hence coincides with (Xmod,S0,S1) (up to a ﬁnite cover) in arbitrarily small tubular neighborhoods of each component of ∂B0. Lemma 6.3 states the existence of a diffeomorphism f of R2 S1 that: mod × Preserves each X -orbit; more precisely, for every point p there is t • mod ∈ [ 1 ,3] such that f (p) X (t,p); 5 mod = mod Coincides with f for large radius x2 y2 (so we can easily replace f • ext + ext with f in tubular neighborhoods of ∂B ); mod p 0 JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 12 CHRISTIAN BONATTI AND NANCY GUELMAN

Possesses two disjoint regions U and V , (coinciding with U and • mod mod ext Vext for large radius) such that Umod is an attracting region for fmod, and 1 Vmod is an attracting region for fmod− ; Possesses a derivative in the direction of Xmod that is uniformly contracting • 1 in Umod and uniformly expanding in fmod− (Vmod); Represents a Morse Smale diffeomorphism having exactly 4 fixed points • when restricted to the circle (0,0) S1, where two of the fixed points are in × Umod and the other two are in Vmod; Features other conditions that are useful in ensuring the partial hyperbol- • icity of the final map.

Proposition 6.9 then veriﬁes that by replacing fext with (lifts on the ﬁnite cover of) fmod in tubular neighborhoods of each component of ∂B0, one obtains an Axiom A + strong transversality and partially hyperbolic diffeomorphism f of M of the form p X (t(p),p), proving most of Theorem 1.3. (It remains to show 7→ that f has only two basic pieces, an attractor and a repeller; this is done in the last section.) All that remains is to prove Lemma 6.3; that is, to construct fmod. For this construction there are two broad steps: Enlist a geometric trick to replace the non-disjoint sections S and S by • 0 1 disjoint sections Σ0 and Σ1, thereby allowing us to build the regions Umod and Vmod; Construct the diffeomorphism f itself. This construction will be done • mod in 3 steps, as outlined subsequently.

2.4. The local model: cutting the helicoids for getting disjoint sections. For the model vector field Xmod, the stable manifold of the periodic orbit is the annulus {x 0} R2 S1 and the unstable manifold is the annulus {y 0}. = ⊂ × = The intersection of the half-helicoid S with the half spaces {x 0} and {x 0} 0 ≤ ≥ are half-planes S0+ and S0−, whose boundaries consist in a union of two half-lines parallel to the y-axes and a Xmod-orbit segment contained in the periodic orbit. One makes an isotopy in S0+ and S0− along the orbits of the vector field supported in a tubular neighborhood of the periodic orbit in such a way that one gets two Σ Σ Σ Σ smooth half-plane 0+ and 0−. As 0+ and 0− coincide with S0+ and S0− outside of a neighborhood of the periodic orbit, their boundaries coincide outside of this neighborhood. We define Σ Σ+ Σ− (See figures 3 and 4). It is diffeomorphic 0 = 0 ∪ 0 to the annulus obtained by cutting a plane R2 along a compact segment. One builds Σ Σ− Σ+ in the same way by pushing S− S {y 0} and 1 = 1 ∪ 1 1 = 1 ∩ ≤ S+ S {y 0} by an isotopy along the X orbit. 1 = 1 ∩ ≤ mod The surprising fact, which is not readily obvious and which represents the starting point of this work, is that one can build Σ0 and Σ1 so that they are disjoint (see figure 4). The regions Umod and Vmod will be small neighborhoods of Σ0 and Σ1.

2.5. The local model: in the quadrants. That is the straightforward part of the construction. The stable and unstable manifold of the periodic orbit of Xmod

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2 1 split R S in 4 quadrants C ±±, { , }, corresponding to the signals of the x × ±Σ ∈ + − and y coordinates. The half-plane 0− induces a transverse cross-section of Xmod Σ in the two quadrants C −− and C −+, and 0+ induces a transverse cross-section Σ Σ of Xmod in the two remaining quadrants. In the same way, 1− and 1+ induce Σ Σ cross-sections of Xmod in all quadrants (2 for 0− and 2 for 0+). Hence, each quadrant is endowed with two complete cross-sections, one contained in Σ and the other in Σ . One builds diffeomorphisms f ±±, { , } by 0 1 ± ∈ + − repeating the procedure for a suspension ﬂow:

Diffeomorphism f ±± preserves each X -orbit; that is, each map is of the • mod form p Xmod(t(p),p), with t(p) 0; 7→ > , Both cross-sections are invariant, and f ± ± coincides on each of them with • the first-return map of the Xmod-orbits; The union of the X -orbit segments of length 2δ centered at any point of • mod the cross-section in Σ0 will be an attracting region; The union of the X -orbit segments of length 2δ centered at any point of • mod the cross-section in Σ1 will be a repelling region. The diffeomorphisms f ±±, { , }, coincide with the expression of f far ± ∈ + − ext from the periodic orbit. The difficulty is that the diffeomorphisms f ±±, { , } ± ∈ + − do not coincide with each other on the boundary of the quadrants. It remains to glue together these four diffeomorphisms, preserving the attracting and repelling regions and Xmod-orbits. We solve this difficulty in the next two steps. 2.6. The local model: in a small neighborhood of the axis. The four quadrants intersect along the periodic orbit of Xmod. We first glue together the diffeomorphisms f ±±, { , } in a neighborhood of the periodic orbit. ± ∈ + − Σ Σ Σ Σ This is also an easy step because any half-plane 0−, 0+, 1− and 1+ meets the periodic orbit (0,0) S1 at just one point, the four points are disjoint and × alternate (for the cyclic order on the periodic orbit) on Σ0 and Σ1. Hence, the attracting and repelling regions in the four quadrants are pairwise disjoint along (0,0) S1, and induce alternately attracting and repelling segments. × Σ Σ Σ Σ Furthermore, we have built 0+, 0−, 1+ and 1− in such a way that in a neighborhood of the periodic orbits, they are contained in horizontal planes (i.e., parallel to R2 {0}). × We build a diffeomorphism f in a neighborhood of (0,0) S1 that coincides 0 × with each of the f ±± on their corresponding attracting or repelling regions, and that induces on (0,0) S1 a Morse–Smale diffeomorphism having four fixed points, × 1 which are the intersections of (0,0) S with Σ0 and Σ1. × , We then glue each diffeomorphism f ± ± with f0 by a bump function defined 1 , on the neighborhood of (0,0) S . The resulting diffeomorphisms f ± ± of the × 0 quadrants coincide in a neighborhood of the periodic orbit (because they coincide with f0). Two of these diffeomorphisms on adjacent quadrants also coincide (on the common boundary of the quadrants) far from the periodic orbit. But there remains a bounded region in the intersection of two adjacent quadrants where the diffeomorphisms do not coincide. This is the last step of our construction

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2.7. The local model: on the walls. Consider now two adjacent quadrants, for instance C +− and C ++. Their intersection is the half cylinder {x 0, y 0}. This Σ 1 1 1 ≥ = cylinder meets 1+ along the straight line z 2 2 S . The diffeomorphisms , , = =− ∈ f0+ − and f0+ + coincide on the 2δ-Xmod-orbit segment centered at the points of Σ this straight line. 1− intersects this half-plane just at the point (0,0,0), and the , Σ diffeomorphisms f0± ± coincide in the neighborhood of that point. However, 0+ Σ and 0− induce on this half-plane two different curves; these two curves are disjoint in a neighborhood of the periodic orbit and coincide far from the periodic orbit. The union of the 2δ-Xmod-orbit segments centered at points on one of , these curves is an attracting region for f0+ + and the corresponding region for the , , other curve is an attracting region for f0+ −. Gluing the diffeomorphisms f0+ + , and f0+ − without breaking the attracting regions is the main technical difﬁculty of our construction. The reader can get an intuitive understanding of our solution by looking at Figures 8 and 9.

2.8. Gluing diffeomorphisms. Several times in our construction we must glue two different locally deﬁned diffeomorphisms f and g that preserve every X - orbit. More precisely, they are of the form p X (p,t (p)) and p X (p,t (p)), 7→ f 7→ g where t f (p) and tp (g) are strictly positive numbers depending smoothly on p. We will use bump functions for gluing f to g. This leads to some difﬁculties:

Why is the map obtained by gluing two diffeomorphisms a diffeomorphism? • Why does it preserve the central foliation? • Solving the second difficulty is easy: we will glue f and g by considering barycentrical combination of t f (p) and tg (p), the coefficient of the barycenter being given by a bump function. However, the derivative of the bump function in the direction of X induces a difficulty since the resulting is a diffeomorphism. For solving this difficulty, we will only glue local diffeomorphisms f and g that are related in a particular way: The diffeomorphisms f and g are defined on a region U that is the union of a smooth family {Ip }p S of pairwise disjoint orbit segments, with initial points p ∈ in a cross-section S. For any p S, the image of I under f and g is the same ∈ p orbit segment: f (I ) g(I ). Moreover, f and g coincide on the two extremal p = p subsegments of Ip of a fixed length ε. Let ψ: S [0,1] be a smooth function, with ψ(r ) 1 for r close enough to ∂S, → = and ψ(r ) 0, for r far from ∂S. = Consider now the map h defined as follows: given any p S, the restriction ∈ of h to I is the map q X (q,t (q)) with t (q) ψ(p)t (q) (1 ψ(p))t (q) (so p 7→ h h = f + − g the bump function we use is constant on the orbit segment Ip , thus solving the difficulty given by its derivative along the orbits). Then h is a diffeomorphism that coincides with f in a neighborhood of ∂U and with g far from ∂U, and such that h(I ) f (I ) g(I ) for every p S. p = p = p ∈ JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 AXIOM A DIFFEOMORPHISMS DERIVED FROM ANOSOV FLOWS 15

3. BASIC DEFINITIONS AND NOTATIONS The aim of this section is to brieﬂy recall the basic deﬁnitions and properties of hyperbolicity and partial hyperbolicity. In what follows, let f : M M be a diffeomorphism on a compact Riemann- → ian manifold M, and let K M be an invariant compact set. ⊂

3.1. Hyperbolic dynamics. One says that K is hyperbolic if there is an integer n 0 and a D f -invariant splitting TM E s E u of the tangent bundle of M > |K = ⊕ over K into the direct sum of sub-bundles E s and E u, such that for any point x K , any vectors u E s(x), and v E u(x) one has: ∈ ∈ ∈ 1 D f n(u) u and D f n(v) 2 v . k k≤ 2k k k k≥ k k

One says that D f contracts uniformly the vectors in E s and expands uniformly the vectors in E u. The bundles E s and E u are called the stable and unstable bundles of K , respectively. We refer to [21] for a nice survey on elementary properties of hyperbolic sets. One of the main properties of a hyperbolic set K is the existence of stable and unstable manifolds through each point of K . The stable manifold W s(x) of a point x K is the set of points y for which the distance d(f n(y), f n(x)) tends ∈ to 0 for n ; for ε 0, the local stable manifold W s(x) is the set of points → +∞ > ε whose distance d(f n(y), f n(x)) remains smaller than ε for n 0. For ε 0 small s ≥ > enough, the local stable manifolds {Wε (x)}x K form a continuous family of dis- ∈ joint embedded C 1-disks centered at x and tangent at x to the stable space E s(x). Furthermore W s(x) W s(x); one deduces that W s(x) isa C 1 injective immer- ε ⊂ sion of E s(x). One defines the unstable (and local unstable) manifold of x as 1 its stable (local stable) manifold for f − . The stable (resp. unstable) manifold W s(K ) (resp. W u(K )) of the hyperbolic set K is the union of the stable (resp. unstable) manifold of the points x K , and (according to the shadowing lemma) is ∈ the set of points whose ω-limit set (resp. α-limit set) is contained in K . A basic set is a transitive hyperbolic set K admitting an isolating neighborhood U; this means that K is the maximal invariant set in U (in formula K n = n Z f (U)). A basic set K is a hyperbolic attractor if it admits an isolating neighborhood∈ U that is strictly positively invariant: the image of the closure of U is T contained in the interior of U; in this case, W u(K ) K . A hyperbolic repeller is a 1 s = hyperbolic attractor for f − and it holds that W (K ) K . = A diffeomorphism f satisfies Axiom A if the non-wandering set Ω(f ) is the closure of the periodic points of f and it is hyperbolic. One important property of an Axiom A diffeomorphism f is that the non-wandering set is the disjoint union of finitely many basic sets called the basic pieces of f . A diffeomorphism f satisfies the strong transversality condition if the stable manifold W s(x) and the unstable manifold W u(y) are transverse for any pair

JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 16 CHRISTIAN BONATTI AND NANCY GUELMAN x, y Ω(f ). One of the main theorems on hyperbolic dynamics (proved by Rob- ∈ bin, Robinson and Mañé) states that f satisfies Axiom A and the strong transversality condition if and only if f is structurally stable: Every diffeomorphism in a small C 1-neighborhood of f is conjugate to f . A cycle of an Axiom A diffeomorphism f is a sequence K1,..Kℓ of distinct basic u s u s pieces such that W (Ki ) W (Ki 1) and W (Kℓ) W (K1) . An Axiom ∩ + 6= ; ∩ 6= ; A diffeomorphism satisfies the no cycle condition if it has no cycles. According to [23], the hyperbolicity of the chain recurrent set is equivalent to Axiom A plus the no cycle condition. 3.2. Partial hyperbolicity. One says that K is partially hyperbolic4, if there is a D f -invariant splitting TM E s E c E u of the tangent bundle of M over K |K = ⊕ ⊕ in direct sum of sub-bundles E s,E c , and E u such that the vectors of E s are uniformly contracted, the vectors in E u are uniformly expanded, and the splitting E s E c E u is a dominated splitting: there is n 0 such that for any point x K ⊕ ⊕ > ∈ and any unit vector u E s(x), v E c (x), and w E u(x) one has ∈ ∈ ∈ 1 2 D f n(u) D f n(v) D f n(w) . k k≤k k≤ 2k k The bundles E s, E c , and E u are called strong stable, central and strong unstable bundles of K , respectively. (We refer to [3, Appendix B] for more precise definitions and basic properties of partial hyperbolicity and dominated splitting). One says that f is partially hyperbolic when the whole manifold M is a partially hyperbolic set. The partial hyperbolicity is an open property for the C r - topology, r 1, on the space of diffeomorphisms. ≥ If f is a partially hyperbolic diffeomorphism then there are foliations F s and F u, called strong stable and strong unstable foliations, tangent to the bundles E s and E u, respectively. These foliations are unique, hence are f -invariant. There is a recent example showing that E c is not always integrable [29]. The existence of a foliation tangent to E c remains an open question in the transitive case with dimE c 1. = 3.3. Hyperbolic set of partially hyperbolic diffeomorphisms. Assume that f is a partially hyperbolic diffeomorphism such that the dimension of the central bundle is 1: TM E s E c E u and dimE c 1. = ⊕ ⊕ = Consider a hyperbolic set K of f . Then the central bundle E c is either expanding or contracting; hence the stable manifold (resp. unstable manifold) of the points x K are everywhere tangent either to E s or to E s E c (resp. either to ∈ ⊕ E c E u or to E u). ⊕ As a consequence, if f is partially hyperbolic with central direction of dimension 1 and furthermore f satisfies Axiom A, then f satisfies the strong transversality condition if and only if for every pair K1,K2 of basic sets one has W s(K ) W u(K ) dim(W s(K )) dim(W u(K )) dim(M). 1 ∩ 2 6= ; =⇒ 1 + 2 ≥ 4According to the different authors, one requires that either both bundles or at least one of the bundles E s and Eu are not trivial (i.e., of dimension 0). Here we will always assume that both > bundles E s and Eu are not trivial.

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Let us state a direct consequence that will be useful in our case:

REMARK 3.1. Let f be a partially hyperbolic diffeomorphism that satisﬁes Axiom A. Assume that every basic piece of f is either an attractor or repeller. Assume ﬁnally that the central bundle is contracted on the attractors and expanded on the repellers. Then f is structurally stable. We have proved in [4, Lemma 3.1] that if A is an attracting basic set of an Axiom A diffeomorphism f E (X ), then the center direction is contracting on ∈ A ; conversely, the center direction is expanding along the repellers of f . With Remark 3.1 above this implies directly:

LEMMA 3.2. Let X be a transitive Anosov ﬂow of a compact 3-manifold. Let f ∈ E (X ) be an Axiom A diffeomorphism such that every basic piece of f is an attractor or repeller. Then f satisﬁes the strong transversality condition and is therefore structurally stable.

4. TOPOLOGICALEQUIVALENCEOF BIRKHOFF SECTIONS 4.1. Birkhoff section of an Anosov ﬂow. Let X be a transitive Anosov vector ﬁeld on a closed 3-manifold M. According to [11] and [7], a Birkhoff section B of X is a compact surface with boundary, embedded in M, with the following properties: Every connected component of the boundary ∂B is a periodic orbit of X ; • The interior of B is transverse to X ; • There is T 0 such that every orbit segment of X of length T meets B. • > In [11] Fried proves the following:

THEOREM 4.1. Every transitive Anosov ﬂow on a closed 3-manifold has a Birkhoff section. By construction, the Birkhoff sections built by Fried satisfy an additional property that we will use in our construction. Let γ denote a connected component of the boundary ∂B, where B is a Birkhoff section built by Fried. Then the bundles E s and E u are orientable along γ; in the sense that the eigenvalues of the derivative of the Poincaré return map associated to γ are positive. In what follows, when we refer to a “Birkhoff section B,” we will always assume that the restrictions to ∂B of the bundles E s and E u are orientable.

4.2. The tame property. The Birkhoff sections B built by Fried satisfy another property that we will use in our construction which we will call the tame prop- s u erty. Let γ be a connected component of ∂B and Wloc (γ) and Wloc (γ) denote the local stable and unstable manifolds of the periodic orbit γ.

DEFINITION 4.2. We say that B has the tame property at γ if there is a neighborhood B of γ in B such that B W s (γ) and B W u (γ) consist in the union γ γ ∩ loc γ ∩ loc of γ with ﬁnitely many compact segments, each of them intersecting γ exactly at one of its ends. We say that B is a tame Birkhoff section if it is tame at each of its boundary components.

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4.3. Topological and local Birkhoff sections. In this work we will use general- izations of the notion of Birkhoff sections: A topological Birkhoff section is a compact surface with boundary B topologically embedded in M such that The embedding of B in M is regular. That is, B is a sub-manifold of M: at • each point x B there are local coordinates ϕ: U V R3 of M centered ∈ → ⊂ at x such that ϕ(B U) V [0, [ R {0}; ∩ = ∩ +∞ × × Every connected component of the boundary ∂B is a periodic orbit of the • flow of X ; The interior of B is topologically transverse to X ; • There is T 0 such that every orbit segment of X of time length T meets B. • > The restrictions to ∂B of the bundles E s and E u are orientable. • B has the tame property at each of its boundary components (the tame • property is defined exactly in the same way as for smooth Birkhoff sections). Let γ be a periodic orbit of the flow of X such that the restriction of E s and E u along γ are orientable. A local Birkhoff section at γ (resp. topological local Birkhoff section at γ) is an embedding (resp. regular topological embedding) of the annulus [0,1] S1 in M such that: × 1. γ is the image of {0} S1; × 2. The image of (0,1] S1 is transverse (resp. topologically transverse) to X ; × 3. There is T 0 and a neighborhood U of γ such that every orbit segment of > X of time length T contained in U meets B. 4. f has the tame property at γ (where the tame property is defined exactly as in Section 4.2).

4.4. The linking number of a Birkhoff section at a boundary component. Let B : [0,1] S1 M be a local Birkhoff section at a periodic orbit γ. As, by defini- × → tion, the eigenvalues associated to γ are positive, the normal bundle N T M |γ ⊂ γ (consisting of all the vectors in TM at a point of γ that are orthogonal to the vector field X ) is orientable. We fix an orientation of the normal bundle N . |γ Consider the unit normal bundle N N of γ: at each point p of γ we 1,γ ⊂ |γ consider the circle of the unit vectors v Tp (M) that are orthogonal to X (p). The 2 ∈ unit normal bundle N1,γ is a torus T , endowed with a projection on the circle γ. The fibers of the projection are circles called meridians. The chosen orientation on N induces an orientation of the meridians. We denote by a H (N ,Z) |γ ∈ 1 1,γ the homology class of the meridians. As we assumed that the eigenvalues associated to γ are positive, the unit tan- s gent vectors at W (γ) induce two disjoint circles on N1,γ, each of them cutting each meridian in exactly one point. These curves are naturally oriented by γ and are homotopic; hence, they define the same homology class b H (N ,Z) ∈ 1 1,γ called the longitude. Notice that we get the same homology class b if we consider the unstable manifold of γ instead of the stable one. We endow the homology space H1(N1,γ,Z) with the basis (a,b).

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At each point p of γ there is exactly one unit normal vector v(p) tangent to B and entering in B (because γ is a boundary component of B). We consider the induced curve (oriented by γ) on N1,γ, and we denote by γB its homology class. There is n(γ,B) Z such that γ n(γ,B)a b. ∈ B = + DEFINITION 4.3. The number n(γ,B) deﬁned above is called the linking number of B at γ.

Notice that the definition of the linking number n(γ,B) depends on the choice of orientation of the normal bundle along γ. Changing this orientation changes n(γ,B) to n(γ,B). − REMARK 4.4. As any orbit cuts B in bounded time, and as B ∂B is transverse to à the flow, one can verify that n(γ,B) 0. 6= The Definition 4.3 holds for smooth local Birkhoff sections. As we will deal with topological Birkhoff sections we now give a topological version of this definition. Let Γ be a tubular neighborhood of γ. Then H1(Γ γ,Z) is canonically iden- 2 à tified to H1(N1,γ,Z) and hence is isomorphic to Z and endowed with the basis (a,b), where a is the meridian and b the longitude. Let B be a topological local Birkhoff section at γ. Consider a closed curve σ B (Γ γ) that is isotopic to γ in B Γ. The homology class of σ in H (Γ γ,Z) ⊂ ∩ à ∩ 1 à does not depend on the choice of σ and it is of the form n(γ,B)a b; the integer + n(γ,B) is called the linking number of B at γ. This definition coincides with the above definition if B is a smooth local Birkhoff section. We will see in the next sections that the linking number is the only invariant of a local Birkhoff section, up to isotopies, obtained by pushing the Birkhoff section along the orbits of the flow.

4.5. Homological intersection in a neighborhood of γ. Let γ be a normally oriented periodic orbit with positive eigenvalues and Γ be a tubular neighborhood of γ. As in the previous section we endow H (Γ γ,Z) with a basis {a,b}, where 1 à a is a meridian and b is a longitude. The choice of the basis {a,b} induces an isomorphism of H (Γ γ,Z) onto Z2. We can also identify H (Γ γ,Z) with the 1 à 1 à homology group of a torus (boundary of a tubular neighborhood of γ). This iden- tiﬁcation allows us to endow H (Γ γ,Z) with the intersection quadratic form. 1 à Evaluated on the basis {a,b}, this intersection form is given by a a b b 0, · = · = a b 1, and b a 1. In other word, the intersection form is the bilinear anti- · = · =− symmetric form on Z2 associated to the matrix 0 1 . 1 0 µ − ¶ Let B : [0,1] S1 M be a local topological Birkhoff section at γ B({0} S1) × → = × such that the boundary component B({1} S1) is disjoint from Γ. We endow B × with an orientation in such a way that the vector ﬁeld followed by this orientation

JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 20 CHRISTIAN BONATTI AND NANCY GUELMAN defines the orientation of the manifold (or the local orientation we have chosen in a neighborhood of γ). Hence, the orbits intersect B with positive intersection number. Let σ Γ γ be a simple closed curve and [σ] ia jb be its homology class; ⊂ à = + in other words [σ] (i, j) Z2 H (Γ γ,Z). = ∈ ≃ 1 à LEMMA 4.5. If σ is a longitude, that is, [σ] (0,1), then the algebraic intersection = number σ B is n(γ,B) . · | | Proof. Consider an essential curve in B on the boundary of a tubular neighborhood of γ. By the definition of linking number, this curve is in the homology class (n(γ,B),1). Hence, the intersection number with a curve in the homology class (0,1) is n(γ,B). Furthermore the curve in this class cuts B always with the same orientation. For this reason we get the claimed equality, up to a sign: σ B n(γ,B) . It remains to see that σ B 0. For that we choose a represen- | · | = | | · > s tative of the class (0,1) that is the concatenation of an orbit segment in Wloc (γ) and a segment in B W s (γ) ( that is possible by the tame hypothesis), proving ∩ loc that the intersection is positive because the chosen orientation of B implies that the orbits intersect B with positive intersection number. 4.6. Quadrants of a local Birkhoff section. Consider a periodic orbit γ with pos- s u itive eigenvalues, Wloc (γ) and Wloc (γ) the local stable and unstable manifolds of γ. Let Γ be a small tubular neighborhood of γ such that Γ (W s (γ) W s (γ)) à loc ∪ loc has exactly 4 connected components. The choice of a transverse orientation of γ induces a cyclic order on these components. We call these quadrants of Γ, and we denote by Γ , i Z/4Z the closure of these connected components. i ∈ Let D Γ be a small disk transverse to X and cutting γ in a point x . One ⊂ D chooses D in such a way that D (W s (γ) W s (γ)) has exactly 4 connected à loc ∪ loc components, each of them contained in one of the quadrants Γi . We call the closure of these components quadrants of D, and we write D D Γ . i = ∩ i We denote by PD the first-return map on D defined in a neighborhood of xD . Notice that PD respects the quadrants: it induces a homeomorphism from a neighborhood of xD in Di on another neighborhood of xD in Di . Let B Γ be a small local topological Birkhoff section at γ (satisfying the tame ⊂ property at γ). Consider the connected components of B (W s (γ) W u (γ)) à loc ∪ loc containing a point of γ in their closure. The closure of these connected components are called quadrants of B. Any quadrant of B is bounded by a segment of γ and by two segments in B (W s (γ) W u (γ)) (by the tame property, this ∩ loc ∪ loc intersection consists of finitely many segments having one endpoint in γ).

LEMMA 4.6. Let σ B W s (γ) be a connected component of B W s (γ) γ, ⊂ ∩ loc ∩ loc à which is a segment having an endpoint in γ. Let Bσ denote the square obtained by cutting the annulus B along σ. There are a neighborhood V of γ in Bσ and a continuous and bounded function t : V R such that Π : x X (t (x),x) is a σ → σ 7→ σ continuous map from V to D. Proof. Cutting the annulus B [0,1] R/Z by σ, one gets an immersion of the ≃ × square B [0,1] [0,1] in B M such that the image of {0} [0,1] is precisely σ ≃ × ⊂ × JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 AXIOM A DIFFEOMORPHISMS DERIVED FROM ANOSOV FLOWS 21

γ. Consider a tubular neighborhood Γ D2 S1 of γ such that D D2 {0} and ≃ × = × γ {0} S1. Let Γ˜ D2 R be the universal cover of Γ. Then B admits a lift B˜ = × ≃ × σ σ on Γ˜, which is an embedded compact square. In particular, it is contained in a compact cylinder of the form D [ T,T ]. Let D D2 denote the disk of radius × − r ⊂ r centered at 0. There is r such that the orbit of every point in D [ T,T ] cuts r × − D in exactly one point. This induces a continuous projection of B˜ D [ T,T ] σ ∩ r × − to D.

REMARK 4.7. As B γ is transverse to the orbits, one gets that the projection Π à σ deﬁned in Lemma 4.6 is a local homeomorphism outside of γ and of σ (and is a local diffeomorphism if B is a smooth local Birkhoff section). The projection along the ﬂow preserves each stable and unstable separatrix of γ. Hence, Πσ sends quadrants of B to quadrants of D, and it induces a homeomorphism from each quadrant of B r γ to a quadrant of D r {xD }. From this one deduces the following lemma:

LEMMA 4.8. Each quadrant Γ of Γ contains exactly n(γ,B) quadrants of B. i | | Proof. The annulus B contains an essential curve c close to γ and disjoint from s u γ, cutting Wloc (γ) and Wloc (γ) always with the same orientation. The intersec- s u tion of c with any quadrant is a segment joining Wloc (γ) and Wloc (γ). Then the number of quadrants is precisely the intersection number of c with W s (γ) loc ∪ W u (γ). Notice that c induces in homology the class n(γ,B)a b in H (Γ γ,Z) loc + 1 à = H (N ,Z). As W s (γ) W u (γ) consists of 4 separatrices (half cylinders), one 1 1,γ loc ∪ loc gets that this intersection number is 4 n(γ,B) , which completes the proof. | | We denote by B , j Z/4 n(γ,B) Z the quadrants of B in such a way that: j ∈ | | B Γ if j i modulo 4 • j ⊂ i ≡ Write j i 4k; then B j 1 is the quadrant of B in Γi 1 which is adjacent to • = + + + B j , that is B j B j 1 γ ∩ + à 6= ; 4.7. The quadrants and the first-return map. Items (2) and (3) of definition of a local Birkhoff section presented in Section 4.3 implies that the first-return map P of the orbits of X on B γ is well-defined and continuous. More precisely, B à there are neighborhoods UB and VB of γ in B and a continuous and bounded function t : U γ R such that the map P : x X (t (x),x) induces a home- B B à → B 7→ B omorphism from U γ V γ. Furthermore, the orbit segment joining x to B à → B à PB (x) meets B exactly at its ends x and PB (x). The first-return map PB preserves the stable and the unstable manifolds of γ and hence permutes the quadrants of B: for every quadrant Q of B there is a quadrant Q′ of B such that PB induces a homeomorphism from a neighborhood of γ in Q, minus γ, to a neighborhood of γ in Q′, minus γ. This induces a permutation on the set of quadrants. The first-return map P is a local homeomorphism of B γ preserving the B à orientation and inducing a permutation of the quadrants B j . Thus, there is k such that for every j Z/4 n(γ,B) Z, P maps the quadrants B on the quadrants ∈ | | B j B j 4k . + JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 22 CHRISTIAN BONATTI AND NANCY GUELMAN

As any simple essential curve in B disjoint from γ induces in homology the class n(γ,B)a b in H (Γ γ,Z) H (N ,Z), one deduces: + 1 à = 1 1,γ LEMMA 4.9. PB maps B j onto B j 4 if n(γ,B) 0 and onto B j 4 if n(γ,B) 0, for + > − < every j Z/4 n(γ,B) Z. ∈ | | 4.8. Projection of a local Birkhoff section along the flow. As in the previous section, we consider a local topological Birkhoff section, B, at a (normally oriented) periodic orbit γ and a disk, D, transverse to X and cutting γ at a point x . Using the notation of the previous section, we divide B in 4n 4 n(γ,B) D = | | quadrants and D in 4 quadrants. As B is not homotopic to a transverse disk, we cannot project it continuously on D along the orbits of X : if one chooses a projection at a point and one tries to extend it by continuity, one gets another value of the projection upon following an essential curve. More precisely, consider the segment σ B W s (γ) that is ⊂ ∩ loc the intersection of quadrants B1 and B4 n(γ,B) . Let Bσ be the square obtained by | | cutting the annulus B along σ. We have seen in Lemma 4.6 that there is a continuous projection Πσ of Bσ to D defined in the neighborhood of γ of the form Π (x) X (t (x),x), where the time projection t is continuous and bounded. σ = σ σ The aim of this section is to estimate the continuity defect of Πσ at σ. More precisely, Bσ contains two copies of σ so Πσ is bivaluate on σ. We will calculate here the difference of these two functions. Let us write n n(γ,B) . For every k {1,...,4n}, k 4j i with i {1,...,4} = | | ∈ = + ∈ and j {0,...,n 1}, there is a neighborhood U of γ in B such that the restric- ∈ − k k tions Π : U D of Π induces an homeomorphism from U γ onto its im- k k → i σ k à age, which is a punctured neighborhood of xD in Di . By construction, one has Πk (x) Πk 1(x) for x Uk Uk 1 and for every k {1,...,4n 1}. = + ∈ ∩ + ∈ − LEMMA 4.10. 1. For x U U one has ∈ 4n ∩ 1 P (Π (x)) if n(γ,B) 0, Π D 1 4n(x) 1 < = (P − (Π1(x)) if n(γ,B) 0 D > 2. Π induces a conjugacy between P n and P :if x U and P n(x) is defined k B D ∈ k B and belongs to U then Π (P n(x)) P (Π (x)). k k B = D k Proof. Let us now prove item (1) above. Let x be a point of B B γ. Let 4n ∩ 1 à α: [0,4n] B be an essential closed curve obtained by concatenating curves → αk :[k 1,k] Bk , k {1,...,4n} with α1(0) x α4n(1) and αk (1) αk 1(0) − → ∈ = = = + ∈ Bk Bk 1. ∩ + Now the projection Π α induces a continuous projection of α on D {x } k ◦ k à D along the orbits of X , whose image is a continuous path Πα: [0,4n] D {x } → à D joining Π1(x) to Π4n(x). The closed path obtained by concatenating the orbit segment joining x to Π1(x), the segment Πα, and the orbit segment joining Π4n(x) to x is homotopic to α in Γ γ. In particular, its homology class in H (Γ γ) is à 1 à (n,1) if n(γ,B) 0 and (n, 1) is n(γ,B) 0. > − < JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 AXIOM A DIFFEOMORPHISMS DERIVED FROM ANOSOV FLOWS 23

Let β D D be a segment joining Πα(4n) Π (x) to Πα(0) Π (x). ⊂ 1 ∩ 4 = 4n = 1 The closed path obtained by concatenating Πα and β is homologous to (n,0) in H (Γ γ,Z). 1 à As a consequence, the closed path obtained by concatenating β and the X - orbit segment joining Π (x) to Π (x) is homologous to (0, 1) if n(γ,B) 0 and 1 4n − > (0,1) is n(γ,B) 0. This proves item (1) of the lemma. < n In proving item (2), one considers a segment δ joining PB (x) to x in Uk . Then the closed path obtained by concatenation of the orbit segment joining x to n PB (x) with δ is isotopic to γ in the tubular neighborhood of γ. Now this segment is isotopic (along the orbits of the ﬂow) to the closed path obtained by Π Π n concatenating the orbit segment joining k (x) to k (PB (x)) and the segment Π Π n Π k (δ) (which is joining k (PB (x)) to k (x) in D). This closed path is therefore Π n homotopic to γ in the tubular neighborhood of γ: this implies that k (PB (x)) is the ﬁrst return on D of the orbit starting at Πk (x), thus proving item (2).

4.9. Equivalent Birkhoff sections. We say that two Birkhoff sections, S0 and S1, are X -isotopic if there is a continuous and bounded function t : S ∂S R 0 à 0 → such that p X (p,t(p)) induces a homeomorphism τ from S ∂S to S ∂S . 7→ 0 à 0 1 à 1 REMARK 4.11. If S and S are X -isotopic, then ∂S ∂S ; furthermore, for ev- 0 1 0 = 1 ery connected component γ of the boundary, the linking numbers n(γ,S0) and n(γ,S1) are equal.

Let γ be a periodic orbit. We say that two local Birkhoff sections B0 and B1 at γ are X -isotopic if there are neighborhoods U0 and U1 of γ in B0 and B1, respectively, and a continuous and bounded function t : U γ R such that 0 à → p X (p,t(p)) induces a homeomorphism τ from U γ to U γ. 7→ 0 à 1 à LEMMA 4.12. Two topological local Birkhoff sections B0 and B1 at γ (satisfying the tame hypothesis) are X-isotopic if and only if the linking numbers at γ are equal: n(γ,B ) n(γ,B ). 0 = 1 Furthermore, if τ: B γ B γ is a homeomorphism realizing the X-isotopy, 0à → 1à then it induces a conjugacy between the ﬁrst-return maps on B0 and B1 (on a neighborhood of γ).

Proof. Let Π0 : B (γ σ) D and Π1 : B (γ σ) D be the projections σ 0 à ∪ → σ 1 à ∪ → deﬁned at Lemma 4.6. We deﬁne the following: 1 1 0 τ (Π )− Π : B (γ σ) B (γ σ). = σ ◦ σ 0 à ∪ → 1 à ∪ Then τ is a homeomorphism from B (γ σ), and item (1) of Lemma 4.10 en- 0 à ∪ sures that it is a homeomorphism τ: B γ B γ. By construction, τ is ob- 0 à → 1 à tained by isotopy along the orbits; in other words, there is a continuous function t : B0 γ R such that τ(x) X (t(x),x) for x B0 γ. Furthermore, t is bounded à → = ∈ à Π0 because the orbits segments joining a point x U0 (reps. x U1) to σ(x) (resp. Π1 ∈ ∈ σ(x)) are uniformly bounded. Hence, we have proven that B0 and B1 are X - isotopic. It remains to prove that τ induces a conjugacy of the return maps. We

JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 24 CHRISTIAN BONATTI AND NANCY GUELMAN write the proof assuming that n(γ,B ) 0; the proof in case n(γ,B ) 0 is iden- 0 > 0 < tical. Let us denote by P P and P P the first-return maps on B and B and 0 = B0 1 = B1 0 1 by t : B γ R and t : B γ R the corresponding return times. Consider 0 0 à → 1 1 à → now a point x B0,k . We have seen that P0(x) belongs to B0,k 4. Let α be a ∈ + segment joining P0(x) to x and contained in B0,k B0,k 1 B0,k 4 (in the case ∪ + ∪···∪ + that n(B ,γ) 1 this is not enough to fix the homotopy class of α; in that case, 0 = we consider a lift of the B0,k to the universal cover of a tubular neighborhood of γ). Consider the closed path β0 obtained by concatenating the orbit segment X ([0,t (x)],x) with α. Then the homology class of β is ( 1,0). 0 0 − Consider now the segment τ(α) joining τ(x) B1,k to τ(P0(x)) B1,k 4 and ∈ ∈ + contained in B1,k B1,k 1 B1,k 4 (once more if n(B0,γ) 1 we need to pass ∪ + ∪···∪ + = to the universal cover of a neighborhood of γ). The orbit segment X ([0, t(x) − + t(P (x)) t (x)],τ(x)) is joining τ(x) to τ(P (x)). 0 + 0 0 Finally, consider the closed path β1 obtained by concatenating the segments X ([0, t(x) t(P (x)) t (x)],τ(x)) and τ(α). The segment β is isotopic to β − + 0 + 0 1 0 along the orbits so that its homology class is ( 1,0). This implies that the inter- − section number β B is 1 and therefore that τ(P (x)) is the first return map of 1 · 1 B0 τ(x) on B1. This completes the proof. Lemma 4.12 allows us to prove the following:

LEMMA 4.13. Let B be a topological Birkhoff section of X and B be a local topological section at a component γ of ∂B. Assume that n(γ,B) n(γ,B). Then for = any neighborhood O of γ there is a topological Birkhoff sectione of X that is X- isotopic to B, coincides with B outside of O, and further coincidese with B ina small neighborhood of γ. e Sketch of proof. According to Lemma 4.12 we can push B on B along the orbits of X in a small neighborhood of γ; the time function of this projection is bounded in absolute value by some constant K . Multiplying this time functione by a bump function, ϕ, we get a surface Bϕ immersed in M, transverse to the orbits of the ﬂow, that coincides with B outside of an arbitrarily small neighborhood of γ (contained in O) and with B in a smaller neighborhood of γ. However, the surface Bϕ may not be embedded in M: it may, in fact, have self- intersections. The surface coincidese with B outside of an arbitrarily small neighborhood of γ and with B˜ in a smaller neighborhood of γ; moreover, this surface is obtained by pushing B along the ﬂow with a time bounded by K . Hence we can assume that this surface is embedded outside of O and in a small neighborhood of γ.

CLAIM 4.14. One can choose ϕ such that Bϕ has no self-intersection point in W s (γ) W u (γ) loc ∪ loc s Proof. Consider the intersection of B with Wloc (γ). By the tame property, it consists of 2 n(B,γ) segments ( n(B,γ) in each separatrix) having exactly one end- | | | | point on γ, and their interiors are pairwise disjoint: hence, the segments contained in one separatrix have a natural cyclic order: the ﬁrst-return map on B

JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 AXIOM A DIFFEOMORPHISMS DERIVED FROM ANOSOV FLOWS 25 sends a segment to the next segment for this order; likewise for the segments of B˜ W s (γ). Furthermore, as τ is conjugating the first-return maps on B and B˜, ∩ loc one gets that τ preserves the cyclic order on the set of segments. Yet, we need something more precise. Consider the universal cover of a tubular neighborhood of γ. As τ is isotopic to the identity along the orbits, this isotopy defines a lift of τ sending a lift of B over a lift on B˜. The fact that τ conjugates the first-return maps on B and B˜ implies that the lift of τ preserves the natural order on the components of intersection ˜ s u of the lifts of B and B with the lifts of the separatrices of Wloc (γ) and Wloc (γ). This allows us to choose the bump function in such a way that Bϕ has no self- intersection on W s (γ) W u (γ). loc ∪ loc A small perturbation of ϕ (obtained by pushing Bϕ along the orbits) in the neighborhood of the intersection points allows us to assume that the self- intersections of Bϕ are all transverse and hence consist of finitely many compact curves. As these curves are disjoint from W s (γ) W u (γ), each of them is loc ∪ loc contained in a quadrant. Consider the lift of Bϕ to the universal cover Γ˜ of the tubular neighborhood Γ D2 S1 of γ. A quadrant of B is obtained by pushing a quadrant of B along ≃ × ϕ the orbits by a bounded time and on an arbitrarily small neighborhood of γ; furthermore, in the lift, the orbits of the flow intersect a quadrant of B in at most one point. As a consequence, one gets that each lift of each quadrant of Bϕ is embedded in the universal cover. In each quadrant Γ˜ , j {1,2,3,4} of Γ˜, the quadrants of the lift of B are nat- j ∈ ϕ urally ordered. Let us index them by B n , n Z; with this notation B n and ϕ,j ∈ ϕ,j n kn(γ,B) B + , k Z, are lifts of the same quadrant of B . ϕ,j ∈ ϕ Let k : max{n m B n B m } and ℓ the sum of the number of con- ϕ = − | ϕ,j ∩ ϕ,j 6= ; ϕ m k nected components of B + ϕ B m for m 1,...n(γ,B). Let us assume that ϕ ϕ,j ∩ ϕ,j = has been chosen in such a way that kϕ is the minimum possible on all the ϕ; let k0 denote this minimum. We assume also that ℓϕ is the minimum possible for all ϕ, with k k . ϕ = 0 One concludes the proof of Lemma 4.13 by proving:

CLAIM 4.15. Assume that k 0 and ℓ 0. Then there is ϕ˜ with k ˜ k or ϕ > ϕ > ϕ < ϕ k ˜ k and ℓ ˜ ℓ . ϕ = ϕ ϕ < ϕ Proof. By deﬁnition of k , there are m, n m k , a point x B m and t 0 ϕ = + ϕ ∈ ϕ,j > such that X ( t,x) B n . Consider the set − ∈ ϕ,j ∆ {x B m t(x) 0, X ( t(x),x) B n }. = ∈ ϕ,j |∃ ≥ − ∈ ϕ,j It is a compact set in B m bounded by B m B n , which consist in ﬁnitely many ϕ,j ϕ,j ∩ ϕ,j circles. Notice that t(x) is unique (because the orbits intersect the quadrants in at most one point). Let D be a connected component of this set, and let D m n = {X ( t(x),x),x D }. Notice that the segments of orbits joining X ( t(x),x) to x, − ∈ m − for x D form a continuous family of segments. This allows to make an isotopy ∈ m JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 26 CHRISTIAN BONATTI AND NANCY GUELMAN

m of Bϕ,j by pushing along the negative orbits the points in a small neighborhood of D in B m ; by this isotopy, any point in D is transformed in a point y m ϕ,j m = X ( t,x) with t t(x). Hence, the negative orbit of the new point (in the new m− > n ∆ Bϕ,j ) is now disjoint from Bϕ,j : we removed the component Dm of . There are two difficulties to be solved: First, one needs to show that this isotopy can be obtained by a choice of ϕ; second, one needs to show that this isotopy either decreases kϕ or keeps kϕ identical but decreases ℓϕ. For solving the first difficulty, one needs to show that one can change ϕ on Dm without changing ϕ on D . If k n m is not a multiple of n(γ,B), B n B m are n ϕ = − ϕ,j ϕ,j obtained by pushing (along the orbits) different quadrants of B. Hence one can change the value of ϕ independently on these two quadrants. If kϕ is a multiple of n(γ,B) we have to show that Dn and Dm are coming from disjoint regions of the same quadrant. Arguing by contradiction, we assume that there is a point y in Dn that is the image under the cover automorphism

h : D2 R D2 R,(p,t) (p,t k ) kϕ × → × 7→ + ϕ of a point x D . In that case there is z D such that X ( t(z),z) y. But ∈ m ∈ m − = X ( t(x), y) is the image under h of X ( t(x),x) B n , hence X ( t(x), y) − kϕ − ∈ ϕ,j − ∈ n k + ϕ Bϕ,j . As a consequence,

n k m 2k X ( t(x) t(z),z) B + ϕ B + ϕ . − − ∈ ϕ,j = ϕ,j

m 2k Therefore B + ϕ B m , leading to a contradiction. ϕ,j ∩ ϕ,j 6= ; For solving the second difficulty, it is enough to see that if x Dm and the i ∈ orbit interval X (( t(x),0],x) cuts Bϕ,j then i m kϕ: these indices are the − m | − |< possible new intersections of Bϕ,j with the other quadrants after modification. First notice that one has i m k n: in fact, if i n and there is x B m ≤ + ϕ = > ∈ ϕ,j and t 0 such that X ( t,x) B i then B m B i ; the conclusion follows > − ∈ ϕ,j ϕ,j ∩ ϕ,j 6= ; now from the definition of k . On the other hand i n because the orbit seg- ϕ 6= ment X ([ t(x),0],x) would cut B n twice, which is impossible. So i m k . − ϕ,j − < ϕ Analogously, notice that a negative orbit starting at y X ( t,x) B i cut B n . = − ∈ ϕ,j ϕ,j As i n, this implies that B i B n . Hence i m, which concludes the < ϕ,j ∩ ϕ,j 6= ; ≥ proof.

This concludes the proof of Lemma 4.13.

As a consequence of Lemma 4.13 we get:

LEMMA 4.16. Let X be a transitive Anosov ﬂow with oriented center-stable and center-unstable foliations on a closed 3-manifold M. Every topological Birkhoff section B˜ is X-isotopic to a (smooth) Birkhoff section B.

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Proof. Notice that by Lemma 4.13, the topological Birkhoff section B˜ is X -isotopic to a (smooth) Birkhoff section B in a neighborhood of γ. Outside of a neighborhood of the boundary, B˜ is topologically transverse to X , so locally we can perturb B˜ to have a smooth Birkhoff section.

The following straightforward corollary is the main goal of this section:

COROLLARY 4.17. Let X and Y be two transitive Anosov ﬂows with oriented foliations deﬁned on closed 3 manifolds M and N respectively. Assume that X and Y are topologically equivalent by a homeomorphism h : M N. Let B be a Birkhoff → section of X. Then there is a Birkhoff section B ′ of Y that is Y -equivalent to h(B). In particular:

∂B ′ h(∂B); • = For every component γ of ∂B onehasn(γ,B) n(h(γ),B ′). • = 4.10. Foliations induced on a Birkhoff section. Let X be an Anosov vector field on a closed 3-manifold, and let γ be a periodic orbit of X such that the stable and unstable bundles are oriented along γ. Let B be a tame local topological Birkhoff section at γ. Let B denote B γ. Then B is homeomorphic to a punctured disk; 0 à 0 furthermore, it is transverse to X so the weak stable and unstable foliations of X s u induce on B0 a pair of transverse foliations F and F . Let ∆ be the disk obtained by compactifying B0 by adding a point 0 (in other words ∆ is the quotient of the annulus B by the boundary component γ). We endow ∆ with the foliations F s and F u in such a way that 0 is the unique singular point. Exactly as Fried noticed in the case of a global Birkhoff section, there is a neighborhood of 0 in ∆ such that on this neighborhood, the pair of foliations F s and F u are conjugate to a pseudo-Anosov type of singular foliation: each foliation has a saddle type singularity with 2 n(γ,B) separatrices (also called prongs | | in the usual terminology for Pseudo-Anosov maps). The stable prongs (or separatrices) of the singularity correspond to the connected component of the inter- s section of B0 with W (γ). More precisely, in the neighborhood of 0, the pair (F s,F u) is conjugate to the pair of foliations obtained by endowing the unit disk of R2 with the trivial horizontal and vertical foliations and by considering a ramified cover of this disk, with n(γ,B) folds having a unique ramification at 0. | |

5. NORMAL POSITION OF A BIRKHOFF SECTION 5.1. Model of local Birkhoff sections: regular helicoid transverse to a hyperbolic periodic orbit . In this section we choose a simple linear model for a vector field in a neighborhood of a periodic orbit. We then show that regular half- helicoids are local Birkhoff sections for this model vector field. This will provide us model Birkhoff sections of any linking number for the model vector field.

5.1.1. The model vector ﬁeld at a periodic orbit. We denote by S1 the circle R/Z.

JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 28 CHRISTIAN BONATTI AND NANCY GUELMAN

We consider the vector ﬁeld on R2 S1, whose expression in canonical coor- × dinates is ∂ ∂ ∂ Xmod(x, y,z) (log2)x (log2)y . = ∂x − ∂y + ∂z 3 We denote by X˜mod the lift of Xmod to R . Notice that the time-one map, X : R2 S1 R2 S1 mod1 × → × is 1 X (x, y,z) (2x, y,z). mod1 = 2 We ﬁx a Riemaniann metric on R2 S1 such that X 1. × k mod k=

5.1.2. The half-helicoid. We consider two surfaces with boundary S˜0 and S˜1, diffeomorphic to the half-plane R [0, ) and properly embedded in R3 and de- × +∞ ﬁned as follows:

S˜ {(x, y,z) R3 r 0,(x, y) (r cos2πz,r sin2πz)} 0 = ∈ |∃ ≥ = S˜ {(x, y,z) R3 r 0,(x, y) (r cos2πz,r sin2πz)} 1 = ∈ |∃ ≥ =− 1 1 {(x, y,z) R3 r 0,(x, y) (r cos2π(z ),r sin2π(z ))} = ∈ |∃ ≥ = − 2 − 2 The surfaces S˜ and S˜ are two half-helicoid. The union S˜ S˜ is a whole 0 1 0 ∪ 1 helicoid. The intersection S˜ S˜ is the z axis, which is the boundary ∂S˜ ∂S˜ . 0 ∩ 1 0 = 1 We write Int(S˜ ) S˜ ∂S˜ . i = i à i Notice that S˜ and S˜ are invariant under the transformation (x, y,z) (x, y,z 0 1 7→ + 1). Hence they induce two surfaces S and S properly embedded in R2 S1. 0 1 × Finally, notice that the transformation (x, y,z) (x, y,z 1 ) induces a diffeo- 7→ + 2 morphism from S˜0 to S˜1 and from S˜1 to S˜0

5.1.3. The half-helicoids are local Birkhoff sections of the model vector field. No- tice that the surfaces S0,S1 and the vector field Xmod are invariant for the natural action of the maps h : R2 S1 R2 S1 defined as (x, y,z) (αx,αy,z), for α × → × 7→ every α 0. > LEMMA 5.1. For i {0,1}, thesurface Int(S ) is transverse to the vector field X . ∈ i mod

Proof. As the surfaces S0,S1 and the vector ﬁeld Xmod are invariant under the maps h for every α 0, it is enough to show the transversality at any point α > (x, y,z) with x2 y2 1. At this point the horizontal component of X (x, y,z) + = mod is log2 and it is strictly less than the vertical component of Xmod (x, y,z) that is 1. The tangent space at such a point is generated by the radial vector ﬁeld ∂/∂r and ∂/∂z ∂/∂θ. As a consequence, the horizontal component of any vector + tangent to Si at such a point is larger than the vertical component: this proves the transversality.

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5.1.4. Bounds of the return maps.

LEMMA 5.2. Any orbit segment σ of X with length ℓ(σ) 5 intersects S . • mod ≥ 4 i Any orbit segment σ of X with length ℓ(σ) 3 meets S in at most 1 • mod ≤ 4 i point. Any orbit segment σ of X joining S to S has its length larger than 1 . • mod 0 1 4 Proof. Notice that the quadrant C ++ {x 0, y 0} is invariant under X . The = ≥ R≥2 1 1 intersection S0 C ++ is contained in {(x, y,z) S z [0, 4 ]} and S1 C ++ is ∩ R2 1 1 3 ∈ × | ∈ ∩ contained in {(x, y,z) S z [ 2 , 4 ]} ∈ × | ∈ 1 As S and S are complete sections of X in C ++ {(0,0)} S , the bounds 0 1 mod à × in the lemma are now straightforward consequences of the fact that Xmod is a ∂ unitary vector (by definition of our metrics) and its coordinates on ∂z is equal to 1. The proof is analogous in the other quadrants {x 0, y 0}, {x 0, y 0}, and ≥ ≤ ≤ ≥ {x 0, y 0}. ≤ ≤ 5.1.5. Model for local Birkhoff section with an arbitrary linking number. By construction, the linking number of the local Birkhoff sections S0 and S1 at the periodic orbit of Xmod is equal to 1. One gets models for Birkhoff sections with arbitrary linking numbers by considering a finite covering of our model. For every n 0, we consider the n-folds covering R2 R/nZ R2 R/Z. We > × → × denote by X the vector field on R2 R/nZ defined by mod,n × ∂ ∂ ∂ Xmod,n(x, y,z) (log2)x (log2)y = ∂x − ∂y + ∂z

In other words, Xmod,n is the lift of Xmod to R/nZ. We denote by γmod,n the periodic orbit of Xmod,n. and by S0,n and S1,n the lifts to R2 R/nZ of S and S (or equivalently, S are the quotient of S˜ on R3 by the × 0 1 i,n i map (x, y,z) (x, y,z n)). One easily verifies the following: 7→ + LEMMA 5.3. The surfaces S0,n and S1,n are Birkhoff sections of Xmod,n at γmod,n whose linking number is n(γ ,S ) n(γ ,S ) n. mod,n 0,n = mod,n 1,n = We have now a model vector field and a model Birkhoff section corresponding to any positive linking number n. For getting a Birkhoff section with linking number equal to n it is enough to consider the vector field X and the im- − mod,n age of S under the symmetry (x, y,z) (x, y,z). 0,n 7→ − 5.2. Normal forms of local Birkhoff section. In this section, D2 denotes the unit disk of R2. The solid torus D2 R/nZ is endowed with the coordinates (x, y,z). × DEFINITION 5.4. We say that an Anosov flow X on a 3-manifold and a Birkhoff section B of X are in normal form if, for every boundary component γ of ∂B, there is an integer n 0, a neighborhood O of γ, and a diffeomorphism Γ : O > γ γ γ → D2 R/nZ such that: × 2 The vector field Γ ∗(X ) is the restriction of X to D R/nZ. • γ mod,n × JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 30 CHRISTIAN BONATTI AND NANCY GUELMAN

The image under the map Γ of the intersection B O is the half-helicoid • γ ∩ γ S D2 R/nZ. 0,n ∩ × Notice in the deﬁnition above that the image Γγ(γ) is the circle γ {x y 0}. mod,n = = = Proposition 5.5 asserts that every pair (X ,B), where X is an Anosov vector ﬁeld and B is a Birkhoff section of X , can be put in a normal form.

PROPOSITION 5.5. Given any pair (X ,B), where X is a transitive Anosov flow on a closed 3-manifold with oriented foliations and B is a Birkhoff section of X , there is an Anosov vector field Y endowed with a Birkhoff section B such that: Y is topologically equivalent to X via a homeomorphism h : M M (which • → maps the oriented orbits of X to the oriented orbits ofe Y ), The Birkhoff section B is Y -equivalent to the topological Birkhoff section • h(B). In particular, ∂B h(∂B). = (Y ,B) are in normal form.e • e 3 5 The return time r (p) of a point p B γ to B belongs to ( 4 , 4 ) • e ∈ à Proof. For any component γ of ∂B we consider the strictly positive integer n i e e i = n(γ ,B) , modulus of the linking number of B at γ . Then by an isotopy of X | i | i preserving the hyperbolicity, we can get an Anosov vector field Y topologically equivalent to X such that, for any component γi of ∂B, the vector field Y coincides with Xmod,ni in a tubular neighborhood Oγi of h(γi ), where h is a homeomorphism realizing the equivalence between X and Y . Then by Lemma 4.16 and Corollary 4.17, there is a Birkhoff section B0 of Y that is Y -equivalent to h(B). If n(γ ,B) 0, using Lemma 4.13 applied to B and the local section S with i > 0 0,ni n n(γ ,B), we conclude that there ise a Birkhoff section B Y -equivalent to i = i i B0 and coinciding with B0 outside of a small neighborhoode of γi and coinciding with S0,ni in a small neighborhood Vi of γi . Up to shrinkinge Vi we may as- 2 1 2 2 sumee that Vi is of the forme {(x, y,z) D S , x y ri } for some ri 0. ∈ × ni + ≤ > The coordinate change (x, y,z) ( x , y ,z) preserves the expression of the vec- 7→ ri ri p tor field Xmod,ni and the equation of the local Birkhoff section S0,ni ; hence, the new Birkhoff section is in normal form at γi . We may apply inductively this argument to all the boundary components with positive linking number. Let us now explain how to deal with components with negative linking number: The coordinate change (x, y,z) ( x, y,z) preserves the vector field X 7→ − mod,n but changes the section (the new section is symmetric with respect to x 0) = and thus changes the sign of the linking number of a local Birkhoff section at the circle S1 . Hence, up to considering this (orientation-reversing) coordinate ni change, we may also put the Birkhoff section in normal form at any component with negative linking number. As seen in Section 5.1, if p γ, then every orbit segment X (p,[0,t]) con- ∉ mod tained in D2 S1 cuts the local Birkhoff section in at most 1 point if t 3 and in at × ≤ 4 least 1 point if t 5 . Therefore, if p γ, then every orbit segment X (p,[0,t]) ≥ 4 ∉ mod,n contained in D2 R/nZ cuts the local Birkhoff section in at most 1 point if t 3 × ≤ 4 JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 AXIOM A DIFFEOMORPHISMS DERIVED FROM ANOSOV FLOWS 31 and in at least 1 point if t 5 . We have the required bounds of the return times ≥ 4 by a time re-parametrization of the orbits of the vector field Y by multiplying Y with a smooth function that is equal to 1 in a small neighborhood of the boundary components γi of ∂B.

5.3. A pair of parallel Birkhoffe sections in normal position. We say that two Birkhoff sections, B and B , of the same Anosov ﬂow X are parallel if B B 0 1 0 ∩ 1 = ∂B ∂B . Notice that parallel Birkhoff sections are X -isotopic. Furthermore, 0 = 1 for every p B ∂B , the X -orbit segment joining p to the ﬁrst return of the ∈ 0 à 0 orbit of p on B0 meets B1 in exactly one point.

DEFINITION 5.6. Let us denote by X an Anosov flow and by (B0,B1) a pair of parallel Birkhoff sections. We say that the triple (X ,B0,B1) is in normal form if for every boundary component γ of ∂B ∂B there is an integer n 0, a neighbor- 0 = 1 > hood O of γ, and a diffeomorphism Γ : O D2 R/nZ such that: γ γ γ → × The vector field Γ ∗(X ) is the vector field X ; • γ mod,n The image Γ (B O ) is the helicoid S D2 R/nZ; • γ 0 ∩ γ 0,n ∩ × The image Γ (B O ) is the helicoid S D2 R/nZ; • γ 1 ∩ γ 1,n ∩ × The return time r (p) of a point p (B ∂B ) to B ∂B belongs to ( 3 , 5 ), • i ∈ i à i i à i 4 4 for i {0,1}; ∈ The return time r (p) of a point p (B0 B1) ∂B0 to B0 B1 ∂B0 belongs • 1 3 ∈ ∪ à ∪ à to ( 4 , 4 ).

This deﬁnition means that both sections B0 and B1 are in normal form, and furthermore, the local normalizing coordinates at γ ∂B ∂B for B and for ∈ 0 = 1 0 B differ by the translation map (x, y,z) (x, y,z 1 ). 1 7→ + 2 The aim of Section 5 is to show the following:

COROLLARY 5.7. Given any pair (X ,B), where X is a transitive Anosov ﬂow on a closed 3-manifold with oriented foliations and B is a Birkhoff section of X , there is an Anosov vector ﬁeld Y endowed with two parallel Birkhoff sections B0 and B1 such that, Y is topologically equivalent to X via a homeomorphism h : M Me (whiche • → maps the oriented orbits of X on the oriented orbits of Y ); The Birkhoff sections B are Y -equivalent to the topological Birkhoff section • i h(B). In particular, ∂B ∂B h(∂B). 0 = 1 = (Y ,B0,B1) is in normale form. • e e Sketch ofe thee proof. We consider (Y ,B0) in normal form (given by Proposition 5.5) such that the return time belongs to ( 3 , 5 ). We write Σ Y (B , 1 ). Then Σ is a 4 4 1 = 0 2 1 Birkhoff section parallel to B0. Thee Birkhoff section B1 is obtained from Σ1 by pushing Σ1 on S1,n along the orbits of Y in the neighborhoode of any boundary component γ. For everyep far from ∂(B ) ∂(B ) wee have that t(p) 1/2. A 0 = 1 = smooth time-rescaling outside of a small neighborhood of the boundary and not so far from ∂(B ) ∂(B ) allows us to get thee returne time conditions. 0 = 1 JOURNAL OF MODERNe DYNAMICSe VOLUME 4, NO. 1 (2010), 1–63 32 CHRISTIAN BONATTI AND NANCY GUELMAN

6. REDUCTIONOFTHEPROOFTOACONSTRUCTIONOFALOCALMODEL The aim of Section 6 is to give most of the proof of Theorem 1.3 (the existence of only two basic pieces will be proven in Section 9), assuming the existence of a local model of diffeomorphism associated to local Birkhoff sections and a local model of vector field (Xmod,S0,S1) built in Section 5. Corollary 5.7 allows us to start with an Anosov vector field X and two Birkhoff sections, B0 and B1, such that the triple (X ,B0,B1) is in normal form. We will as- sociate to (X ,B ,B ) an Axiom A diffeomorphism f E˜(X ) whose non-wandering 0 1 ∈ set consists of only one hyperbolic attractor and one hyperbolic repeller. In this section we will build f outside of a small neighborhood of the periodic orbits in ∂B0. More precisely, we will build a diffeomorphism fext defined on M ∂B . The advertised diffeomorphism f will be obtained by gluing f with a à 0 ext diffeomorphism fmod that will be built in the next sections. 6.1. Building the diffeomorphism f . We fix λ 1 C 0, the constant asso- ext > > ciated to the hyperbolicity of X : for any p M, any unit vectors u E ss(p), ∈ ∈ v E uu(p) and any t 0 one has: ∈ > 1 t t DXt (u) λ− and DXt (v) Cλ . k k≤ C k k≥ 1 We fix α (1,λ 100 ]. All our construction will depend on a (small) number δ 0 ∈ > whose value will be fixed at the end. During the construction, we will omit the dependence on α and δ. 6.1.1. Building f on orbit segments. In M ∂B , the surfaces B ∂B are com- ext à 0 i à i plete sections cutting the orbits in segments, and we will define fext on these segments. More precisely, we denote by P : B ∂B B ∂B and P : B B0 0 à 0 → 0 à 0 B1 1 à ∂B B ∂B the first-return maps of X to the interior of the Birkhoff sections 1 → 1 à 1 B and B . The diffeomorphism f coincides with P and P on B B . 0 1 ext B0 B1 0 ∪ 1 Every point p M (B B ) belongs to exactly one X -orbit segment I ∈ à 0 ∪ 1 = [q ,q ]c or I [q ,q ]c with q B , q B , and whose interior is disjoint 0 1 = 1 0 0 ∈ 0 1 ∈ 1 from B B . Then f (I) will be the segment J [P (q ),P (q )]c or J 0 ∪ 1 ext = B0 0 B1 1 = [P (q ),P (q )]c , respectively. Let us build f on a segment I [q ,q ]c with B1 1 B0 0 ext = 0 1 q B , q B . 0 ∈ 0 1 ∈ 1 The natural parametrization by the flow of X provides identifications of I and J with segments [0,r ] and [0,s], r 0,s 0. In other words, we consider > > the diffeomorphisms [0,r ] I,t X (q0,t) and [0,s] J,t X (PB0 (q0),t). As → 7→ → 1 37→ (X ,B0,B1) is in normal form, the times r and s belong to [ 4 , 4 ]. One chooses a smooth family of diffeomorphisms Θ : [0,r ] [0,s] (See Fig- r,s → ure 1) such that: 1 1. Θr,s(t) α− t for t [0,δ] = ∈ 1 2. Θr,s(t) s α(r t) for t [r α− δ,r ] = − 1 − ∈ − 3. Θ (t) [α− δ,δ] for every t [δ,r δ] r,s ∈ ∈ − Then one defines f : I J by f (X (q ,t)) X (P (q ),Θ (t)). ext → ext 0 = B0 0 r,s JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 AXIOM A DIFFEOMORPHISMS DERIVED FROM ANOSOV FLOWS 33

s α

s δ −

1 α− δ 1 α−

1 0 δ r δ r α− δ r − −

FIGURE 1. The map Θr,s.

Consider now the case of a segment I [q ,q ]c with q B , q B , and = 1 0 0 ∈ 0 1 ∈ 1 J [P (q ),P (q )]c . In this case we consider the parametrization by the ﬂow = B1 1 B0 0 of X : we consider r 0 and s 0 and diffeomorphisms [0,r ] I,t X (q , t) − > > → 7→ 0 − and [0,s] J,t X (P (q ), t). Then one deﬁnes: → 7→ B0 0 − f : I J by f (X (q , t)) X (P (q ), Θ (t)). ext → ext 0 − = B0 0 − r,s 6.1.2. The attracting and repelling regions for fext. We denote byUext and Vext the union of the orbit segments of X with length 2δ centered at the points of B ∂B 0 à 0 and B ∂B , respectively: 1 à 1

U {X (p,t),p B ∂B ,t [ δ,δ]} ext = ∈ 0 à 0 ∈ − V {X (p,t),p B ∂B ,t [ δ,δ]}. ext = ∈ 1 à 1 ∈ − The sets U and V are disjoint for 0 δ 1 . By the estimation of return ext ext < < 8 times and deﬁnition of the smooth family of maps Θr,s one can easily verify:

LEMMA 6.1. The map f defined above is a diffeomorphism of M ∂B . ext à 0 For every p M ∂B there is t [ 1 ,2] such that f (p) X (p,t). • ∈ à 0 ∈ 4 ext = U is the attracting region of f , and f (U ) is contained in the interior • ext ext ext ext of Uext; 1 V is the repelling region of f ; and f − (V ) is contained in the interior • ext ext ext ext of Vext; 1 if p M (∂B U V ), then f (p) U and f − (p) V . • ∈ à 0 ∪ ext ∪ ext ext ∈ ext ext ∈ ext 6.2. The model diffeomorphism. Our main technical lemma is the construction of a diffeomorphism fmod defined in a neighborhood of a periodic orbit of the vector field X on R2 S1. This diffeomorphism will be used as a model for × JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 34 CHRISTIAN BONATTI AND NANCY GUELMAN the announced diffeomorphism f in the neighborhood of the boundary components of the Birkhoff section. We will use the following definition.

DEFINITION 6.2. Let f be a diffeomorphism leaving invariant each orbit of a D f (X (p)) vector field X . The central derivative of f at the point p is Dc f p . = X (f (p) LEMMA 6.3. There is δ such that for every δ (0,δ ] there is a diffeomorphism 0 ∈ 0 f of R2 S1, and two closed subsets U , V R2 S1 with the following mod × mod mod ⊂ × properties: 1. For every point (x, y,z) there is t [ 1 ,3] such that f ((x, y,z)) X (x, y,z); ∈ 5 mod = t 2. Umod is strictly invariant under fmod: fmod(Umod) Int(Umod); and Vmod is 1 1 ⊂ strictly invariant under f − : f − (V ) Int(V ); mod mod mod ⊂ mod 3. U {(x, y,z), x2 y2 2} and V {(x, y,z), x2 y2 2} coincide mod ∩ + ≥ mod ∩ + ≥ with the union of the orbit segments of the flow of X of length 2δ centered at p p the half-helicoid S0 and S1, respectively; 4. The restriction of f to the periodic orbit (0,0) S1 is a Morse–Smale dif- mod × feomorphism of the circle having exactly four fixed points, two of them are in Umod and two in Vmod; 5. Let p (x, y,z) S be a point such that x2 y2 100. Let I be the or- = ∈ 0 + ≥ p bit segment of X joining (x, y,z) to its first return P (x, y,z) on S . Then p S0 0 fmodIp IP (p). Furthermore the expression of fmod in restriction to the = S0 segment Ip is the same as the expression of fext. More precisely the segment contains a unique point q S1. Let r,s,r ′,s′ 0 such that q Xr (p), ∈ 2 > = PS (p) Xr (q) PS (q) Xs(PS (p)) and P (p) Xs (PS (q)). Then 0 ′ 1 0 S0 ′ 1 = = = Θ For t [0,r ] fmod(Xt (p) Xt ′ (PS0 (p)) with t ′ r,s(t) • ∈ = 2 = For t [0,r ′] fmod(X t (PS (p)) X t (P (p)) with t ′ Θr ,s (t) − 0 − ′ S0 ′ ′ • ∈ c = c 1 = 1 6. The central derivatives D fmod and D fmod− are at most α− in Umod and Vmod respectively; c c 1 1 7. The central derivatives D fmod and D fmod− are precisely α− in

U {(x, y,z), x2 y2 10} mod ∩ + ≥ and in q V {(x, y,z), x2 y2 10} mod ∩ + ≥ respectively. More precisely, every orbitq segment I of length 2δ, centered at a point (x, y,z) S (resp. (x, y,z) S ) such that x2 y2 10, is mapped ∈ 0 ∈ 1 + ≥ by f (resp. by f 1 ) in an afﬁne way on the orbit segment of length α 1δ mod mod− p − 1 centered at PS (x, y,z) (resp. P − (x, y,z)). 0 S1 8. For all α′ α, there is N 0 such that for any n N, for any point p U > > ≥ ∈ mod and p′ V and one has ∈ mod c n n n c n n n D f (p) (α′− ,α− ] and D f − (p′) (α′− ,α− ]. mod ∈ mod ∈ DEFINITION 6.4. For every n N 0 we denote by f the diffeomorphism of ∈ à mod,n R2 R/nZ obtained as follows: × JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 AXIOM A DIFFEOMORPHISMS DERIVED FROM ANOSOV FLOWS 35

Let R2 R/nZ R2 S1 be the canonical covering. Then f is the unique lift × → × mod,n of fmod that preserves each orbit of the vector ﬁeld X .

6.3. Gluing the local models with fext. Let (X ,B0,B1) be a transitive Anosov vector field endowed with a pair of parallel Birkhoff sections in normal form given by Corollary 5.7 (and verifying the condition on the return times given by Defi- nition 5.6). For every periodic orbit in ∂B we fix normalizing coordinates Γ: O D2 0 0,γ → × R/nZ (where n is the absolute value of the linking number of γ). We write Oγ 1 2 1 = Γ− (D 1 R/nZ), where D 1 D is the disk of radius centered at 0. 8 × 8 ⊂ 8 For every periodic orbit γ ∂B0 and every r 1 we denote by fr,γ : Oγ O0,γ 1 1⊂ > → the diffeomorphism Γ− h− f h Γ where h : (x, y,z) (r x,ry,z). ◦ r ◦ mod,n ◦ r ◦ r 7→ 2 REMARK 6.5. The map fr,γ is well-defined because if (x, y,z) R R/nZ satisfies 2 2 1 1 2 ∈ × x y , then h− f h (x, y,z) belongs to D R/nZ. + ≤ 8 r ◦ mod,n ◦ r × Proof.p Notice that the expansion of the flow Xmod on the x, y-coordinates at time t is bounded by 2t .

For every point p B ∂B we denote by I the orbit segment of X joining p ∈ 0 à 0 p to its first return PB (p) on B0. By construction, fext(Ip ) IP (p), for every p. 0 = B0 REMARK 6.6. Consider p (B ∂B ) O and (x, y,z) Γ(p). Since every point ∈ 0 à 0 ∩ 0,γ = in I is of the form X (p) with 0 t 5 2, then: p t ≤ ≤ 4 < 1. if x2 y2 1 (that is, p O ) then I O ; + ≤ 8 ∈ γ p ⊂ 0,γ 2. if x2 y2 1 then I O : p + ≤ 32 p ⊂ γ 2 2 1 3. as a consequence, if x y then Ip Oγ and IP (p) O0,γ p + ≤ 32 ⊂ B0 ⊂ LEMMA 6.7. There is r0 1psuch that for every r r0, for every p B0 O0,γ and > 2 2 1 1 ≥ ∈ ∩ (x, y,z) Γ(p) such that x y [ ] one has: fr,γ(Ip ) fext(Ip ) IP (p) = + ∈ 1000 32 = = B0 and the restriction of f and f to this segment are equal. r,γp ext Proof. Take r such that r 1 100 and see definitions of f and f . 0 0 1000 ≫ r,γ ext Now let O (γ) be the union of the orbit segments I for p B such that 2 p ∈ 0 x2 y2 1 , where (x, y,z) Γ(p). + ≤ 16 = pCOROLLARY 6.8. For r r there is a diffeomorphism f : M M that coincides > 0 r → with fext outside of the union of the O2(γ) for all connected component γ of ∂B0 and that coincides with f for x O (γ). r,γ ∈ 2 PROPOSITION 6.9. There is r r such that for every r r the diffeomorphism 1 ≥ 0 ≥ 1 fr satisfies the following properties: 1. f is of the form p X (p), where t(p) is a smooth function with values r 7→ t(p) in [1/5,3]; 2. fr satisfies Axiom A and the strong transversality condition; and 3. fr is partially hyperbolic (its central bundle is generated by X ).

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For proving Theorem 1.3 from Proposition 6.9, it just remains to prove that one can build fr with exactly two basic pieces: a (connected) attractor and a repeller. This will be done in Section 9, using a precise analysis of the dynamics of the diffeomorphisms fr that we have built. Proof. 1. For every r r , the diffeomorphism f is of the form p X (p) with t ≥ 0 r 7→ t ∈ [ 1 ,3]: if p O (γ) then f (p) f , which is in the appropriate form by con- 5 ∉ γ 2 r = ext struction; if p O (γ), then f (p) f (p). Notice that f is conjugate to f S∈ 2 r = r,γ r,γ mod,n by h Γ , which maps the Anosov vector ﬁeld X in M to the linear vector ﬁeld r ◦ γ X in R2 R/nZ. Now the claim comes from the fact that f (h (Γ (p)) mod × mod,n r γ = X (h (Γ (p)) for some t [ 1 ,3]. This concludes the proof of Proposition modt r γ ∈ 5 6.9.1.

2. Let us denote by Ur and Vr the subsets of M deﬁned as follows:

1 2 2 r Ur (hr Γγ)− (Umod {(x, y,z), x y }) Uext Int(Oγ) = ∩ + ≤ 8 ∪ à γ µ ¶ Ã γ ! [ q [

1 2 2 r Vr (hr Γγ)− (Vmod {(x, y,z), x y }) Vext Int(Oγ) = ∩ + ≤ 8 ∪ à γ µ ¶ Ã γ ! [ q [ Then U and V are disjoint compact sets. Moreover, f (U ) Int(U ) (we see r r r r ⊂ r that independently in the parts where fr coincide with fext or with fr,γ using the 1 2 2 r fact that U and (h Γ )− U {(x, y,z), x y } coincides on O ext r γ mod ∩ + ≤ 8 0,γ à Γ 1 2 2 2 ³ 1 ´ γ− ({ x y r }). In the same way fr− (Vr )p Int(Vr ). + ≤ n ⊂ n We denote by Ar n Z fr (Ur ) and Rr n Z fr (Vr ) the maximal invariant p = ∈ = ∈ sets in U and V , respectively. r r T T CLAIM 6.10. The chain recurrent set R(f ) is contained in A R . r r ∪ r Proof. As fr (Ur ) Int(Ur ), there is η 0 such that every η-pseudo orbit (pi )i Z ⊂ > ∈ satisﬁes p U p U , j i. i ∈ r ⇒ j ∈ r ∀ ≥ In the same way, p V p V , k i. i ∈ r ⇒ k ∈ r ∀ ≤ One deduces that R(f ) (U V ) A R (see [8]). r ∩ r ∪ r ⊂ r ∪ r Recall that, by construction, every point p U V satisﬁes f (p) U . ∉ ext ∪ ext ext ∈ ext One deduces that for η 0 small enough, every η-pseudo orbit meeting M > à O meet either U or V . One deduces that every point p R(f ) is either γ γ r r ∈ r contained in A R or there is a boundary component γ of ∂S such that the S r ∪ r 0 orbit of p is contained in Oγ. However, the maximal invariant set of Oγ is γ. Hence, if p R(f ) (A R ), then p belongs to some component γ. However, ∈ à r ∪ r for every point p γ the ω-limit set of p is a periodic point of the restriction ∈ of f to γ, and this periodic point belongs, by construction, to U V , thereby r r ∪ r contradicting the fact that p A R . ∉ r ∪ r JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 AXIOM A DIFFEOMORPHISMS DERIVED FROM ANOSOV FLOWS 37

CLAIM 6.11. The invariant compact sets Ar and Rr are hyperbolic. Furthermore, the stable spaces of the points of Ar (respectively, Rr ) have dimension 2 (resp. dimension 1).

Proof. This is a simple consequence of the fact that fr is of the form p Xt (p) 1 c 1 7→ with t , and the central derivative D f is at most α− on U . Hence every ≥ 5 r r vector in the center-stable direction of the vector ﬁeld X at a point p Ur is n ∈ (uniformly) contracted by D fr for large n 0. Analogously, every vector in the > r n center-unstable direction of X at a point q V is uniformly contracted by D f − ∈ r for large n 0. >

We have proven that the chain recurrent set R(fr ) is hyperbolic, and according to [23], this implies that fr satisﬁes Axiom A and has no cycles. Furthermore, if K and L are two hyperbolic sets of f such that W u(K ) W s(L) then r ∩ 6= ; either K L U • ∪ ⊂ r or K L V • ∪ ⊂ r or K V and L U . • ⊂ r ⊂ r As a consequence one gets that dimW u(K ) dimW s(L) 3 dimM. As we no- + ≥ = ticed in Section 2.3, this implies that fr satisﬁes the strong transversality condition. This completes the proof of Proposition 6.9.2.

n0 3. For all x M, there exist n0,n1 positive numbers such that fr (x) Ur and 1 n1 ∈ ∈ fr− (x) Vr . Let us ﬁx α α′ λ 50 . According to item (8) of Lemma 6.3, there is ∈ < 1< n N 0 such that, in Ur and fr− (Vr ), the central derivative of f , for n N and its > n n ≥ inverse are bounded by α′ λ 50 . < 1 n On the other hand, fr is of the form p Xt (p), with t 5 so D fr contracts s 7→ 1 n ≥ k k the vectors on E by a factor smaller than C − λ− 5 and expands the vectors in u n n E by a factor larger than Cλ 5 . This proves that the contraction of D fr (resp. n ss uu k k D f − ) restricted to E (resp. to E ) is uniformly stronger than the contrac- k r k tion in the direction of the ﬂow; that is, fr is partially hyperbolic and RX is its central bundle.

7. INTHENEIGHBORHOODOFTHEBOUNDARYOFA BIRKHOFF SECTION 7.1. General presentation of our construction. In this section we start the construction of the model diffeomorphism f of R2 S1 announced in Lemma 6.3. mod × For that, we consider R2 S1 endowed with the model vector field X X and × = mod the model Birkhoff section S0 and S1. We divide R2 S1 in regions having global sections cutting the orbits in com- × pact segments in order to define the diffeomorphism segment by segment. The first regions we consider are the quadrant associated to the periodic orbit. As the vector field is the model vector field, the quadrants can be expressed in formula by:

C ++ {(x, y,z),x 0 and y 0}, C +− {(x, y,z),x 0 and y 0}, = ≥ ≥ = ≥ ≤ C −+ {(x, y,z),x 0 and y 0}, and C −− {(x, y,z),x 0 and y 0}. = ≤ ≥ = ≤ ≤ JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 38 CHRISTIAN BONATTI AND NANCY GUELMAN

In order to glue the diffeomorphisms in a quadrant with fext, we will consider transverse sections in each quadrant that coincide with S0 and S1 outside of a neighborhood of the periodic orbit (0,0) S1. We cannot use S and S because × 0 1 these two Birkhoff sections are not disjoint. Section 7.2 will provide us these sections, called Σ0 and Σ1, obtained by pushing S0 and S1 along the orbits in different ways, depending on the quadrants. In each quadrant C ±±, the sections Σ0 and Σ1 induce disjoint smooth surfaces whose boundaries are contained in the boundary of the quadrants. Exactly as we have built fext, we will build diffeomorphisms f ±± of the quadrants, admitting a tubular neighborhood of Σ0 as an attracting region, and a tubular neighborhood of Σ1 as a repelling region. 1 Far from the periodic orbit (0,0) S of X , all the f ±± have the same expres- × sion (analogous to those announced in Lemma 6.3). But these diffeomorphisms 1 f ±± do not coincide on either (0,0) S or its invariant manifolds, which are the × intersections of the adjacent quadrants. For solving this difficulty, Section 7.6 defines a diffeomorphism f0 in the neighborhood of the periodic orbit (once more, f0 is defined on orbit segments obtained by cutting the orbits along local sections through the periodic orbit). A bump function will allow us to glue f0 to the f ±± obtaining new diffeomorphisms f0±± equal to f ±± far from the periodic orbit and equal to f0 in the neighborhood of the periodic orbit.

The most difﬁcult part will consist in gluing the diffeomorphisms f0±± in the intersections of the quadrants. It will be done in Section 8. Let us start by intro- ducing a bump function that we will use many times during this construction. We denote by ψ: [0, ) [0,1] a smooth map such that +∞ → 1 2 The derivative ψ′(x) is strictly negative for x ( , ); • ∈ 3 3 ψ(t) 1 for t [0, 1 ]; and • = ∈ 3 ψ(t) 0 for t 2 . • = ≥ 3

7.2. Surgery on the helicoids S0 and S1 for getting disjoint sections of X . The intersection S− S {y 0} is homeomorphic to a half-plane and dif- 0 = 0 ∩ ≤ feomorphic to [0,1] [0, ): the intersection with the plane y 0 is composed × +∞ = of the half line x 0, y 0,z 1 , the segment x y 0, z [ 1 ,0], and the half ≤ = =− 2 = = ∈ − 2 line x 0, y z 0. ≥ = = Consider (x, y,z) S−. The coordinate z belongs to R/Z; however, for a point ∈ 0 in S0− one has y r sin2πz 0, with r 0. Hence, one may choose a representa- 1 = < > tive z [ ,0]. Now, the map π− deﬁned by ∈ − 2 0

(x, y,z) π0−(x, y,z) X z 1 (x, y,z) 7→ = − − 4 1 is a projection of S− on the half-plane {y 0,z } that is a diffeomorphism in 0 ≤ =− 4 restriction to S− {x y 0}. 0 à = = We will push S0− along the orbit of X in the direction of the projection π0− in order to get a smooth surface that coincides with S− for large radius, i.e r 0 = x2 y2, and that coincides with the half-plane y 0, z 1 for small radius + ≤ =− 4 (see Figure 2). p JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 AXIOM A DIFFEOMORPHISMS DERIVED FROM ANOSOV FLOWS 39

For that we consider a barycenter between (x, y,z) S− and π−(x, y,z) in the ∈ 0 0 orbit segment of X joining these two points and whose coefﬁcient is given by the function ψ deﬁned in Section 7.1. We write

1 2 2 1 Σ− Xt (x, y,z) (x, y,z) S0, y 0,z [ ,0],t ψ( x y ) ( z ) . 0 = ∈ ≤ ∈ −2 = + · − − 4 ½ ¯ ¾ ¯ q ¯ Σ ¯ Notice that 0− is a smooth surface with boundary that coincides with S0− for x2 y2 2 and coincides with the half-plane {y 0,z 1 } for x2 y2 1 ; + ≥ 3 ≤ =− 4 + ≤ 3 1 2 2 1 more precisely, the half disk {y 0,z , x y } is contained in Σ−. p ≤ =− 4 + ≤ 3 p 0 p

-2/3 -1/3 1/3 2/3

S− {y 0} 0 ∩ ≥ S0−

S 0− the disk z 1 = 4

(x,0,0)

1 Π−(x,0, ) 0 − 2 1/4 -2/3 -1/3 1/3 2/3 Π (x,0,0) 1 0− S− {y 0} (x,0, ) 0 ∩ ≥ − 2 -1/3 1/3

Σ 0−

1 FIGURE 2. Pushing S− in the direction of the plane z . 0 =− 4

JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 40 CHRISTIAN BONATTI AND NANCY GUELMAN

In the same way, we push S+ S {y 0} along the orbit of X in the direction 0 = 0 ∩ ≥ of the half-plane y 0, z 1 . One deﬁnes: ≥ = 4

1 2 2 1 Σ+ Xt (x, y,z) (x, y,z) S0, y 0,z [0, ],t ψ( x y ) ( z ) . 0 = ∈ ≥ ∈ 2 = + · − + 4 ½ ¯ ¾ ¯ q ¯ We write Σ Σ− Σ+ (see Figure 3). Notice that 0 = 0 ∪¯ 0

1 1 1 2 2 Σ− Σ+ {y 0,z S ,x } {y 0 z,x }. 0 ∩ 0 = = =−2 = 2 ∈ ≤−3 ∪ = = ≥ 3

Hence, Σ0 is a surface with boundary and corners, whose boundary is contained 2 2 in {y 0,x [ , ]} and it is composed of the segment in ∂Σ− {y 0} join- = ∈ − 3 3 0 ⊂ = ing the points (2/3,0,0) and ( 2/3,0, 1/2), and the segment in ∂Σ+ joining the − − 0 points (2/3,0,0) and ( 2/3,0,1/2) ( 2/3,0, 1/2). − = − −

Σ0

FIGURE 3. The section S0 and Σ0.

In the same way, we deﬁne a surface Σ by pushing S− S {x 0} and S+ 1 1 = 1 ∩ ≤ 1 = S {x 0} along the orbits of X in the direction of the planes {z 0} and {z 1/2} 1∩ ≥ = = respectively. More precisely:

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1 1 For (x, y,z) S− we can choose z [ , ]. We deﬁne: ∈ 1 ∈ − 4 4

1 1 2 2 Σ− Xt (x, y,z) (x, y,z) S1,x 0,z [ , ],t ψ( x y ) ( z) . 1 = ∈ ≤ ∈ −4 4 = + · − ½ ¯ ¾ ¯ q ¯ 1 3 For (x, y,z) S+ we can choose z [ , ]. We deﬁne: ∈ 1 ¯ ∈ 4 4

1 3 2 2 1 Σ+ Xt (x, y,z) (x, y,z) S1,x 0,z [ , ],t ψ( x y ) ( z ) . 1 = ∈ ≥ ∈ 4 4 = + · − + 2 ½ ¯ ¾ ¯ q We write Σ Σ− Σ¯+. Notice that 1 = 1 ∪ ¯1 1 1 2 3 1 2 Σ− Σ+ {x 0,z S , y } {x 0,z , y } 1 ∩ 1 = = = 4 ∈ ≤−3 ∪ = = 4 =−4 ≥ 3 . Hence, Σ1 is a surface with boundary and corners, whose boundary is contained in {x 0, y [ 2 , 2 ]}. Furthermore ∂Σ is composed of the segment in = ∈ − 3 3 1 ∂Σ+ {x 0} joining the points (0, 2/3,1/4) and (0,2/3,3/4) and the segment in 1 ⊂ = − ∂Σ− joining the points (0,2/3, 1/4 3/4) and (0, 2/3,1/4). 1 − = −

LEMMA 7.1. The surfaces Σ0 and Σ1 are disjoint.

Figure 4 shows how surfaces Σ0 and Σ1 ﬁt together, and this may help the reader follow the proof below.

Proof. The proof consists of looking at the z-coordinates of the points in Σ0 and Σ1 according to the signs of the other two coordinates. Figures 5 and 6 may help the reader to follow the proof below. In the quadrant C ++ {x 0, y 0}, it holds that = ≥ ≥ Σ 1 1. (x, y,z) 0+ implies that z [0, 4 ]. Furthermore z 0 y 0 and x 2/3. ∈ Σ ∈ 1 = ⇒ = ≥ 2. (x, y,z) 0− implies that y 0 and z [ 4 ,0]. Furthermore z 0 x 2/3. ∈ = 1 3 ∈ − = ⇒ ≥ 3. (x, y,z) Σ+ implies that z [ , ]. Furthermore z 3/4 1/4 x 0 ∈ 1 ∈ 2 4 = =− ⇒ = and y 2/3. ≥ 1 1 4. (x, y,z) Σ− implies that x 0 and z [ ,0]. Furthermore z y ∈ 1 = ∈ − 4 =− 4 ⇒ ≥ 2/3.

One gets directly that in this quadrant, Σ+ Σ+ . Notice that 0 ∩ 1 = ; Σ+ {x 0, y 0} {z [ 1/4,0]} 1 ∩ ≥ = ∩ ∈ − = ; and hence C ++ Σ+ Σ− . Analogously, ∩ 1 ∩ 0 = ; 1 C ++ Σ+ {x 0} {z [ ,0]} ∩ 0 ∩ = ∩ ∈ −4 = ; so C ++ Σ+ Σ− . In the case that x 0 and y 0, the point (0,0,z) Σ− if ∩ 0 ∩ 1 = ; = = ∈ 0 and only if z 1/4, and (0,0,z) Σ− if and only if z 0, so C ++ Σ− Σ− . =− ∈ 1 = ∩ 1 ∩ 0 = ; We have therefore proved that C ++ Σ Σ . ∩ 1 ∩ 0 = ; The proof in the other quadrants C +−,C −+ and C −− is analogous.

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1/3

Σ0

Σ1

Σ0

FIGURE 4. Σ0 and Σ1 are disjoint.

7.3. First-return maps. Σ Notice that 0− is a global section of the vector field X on the (X -invariant) half Σ Σ space {y 0}. Hence the first-return map P0− : 0− 0− of the orbits of X on the Σ≤ → Σ section 0− is well-defined and it is a diffeomorphism of 0−.

LEMMA 7.2. For every p Σ− the orbit segment joining p to P −(p) has length ∈ 0 0 larger than 1/2.

Proof. That is because Σ− is contained in z [ 1/2,0]. 0 ∈ − Similarly to the above, the first-return maps P + : Σ+ Σ+, P − : Σ− Σ−, and 0 0 → 0 1 1 → 1 P + : Σ+ Σ+ are also well-defined diffeomorphisms. We now consider how the 1 1 → 1 first-return maps P0+ and P0− fit together. Σ Σ Σ Σ 1 LEMMA 7.3. If p 0+ 0− then P0+(p) P0−(p). If q 1+ 1− then (P1+)− (q) 1 ∈ ∩ = ∈ ∩ = (P1−)− (q).

Proof. A point p (x, y,z) belongs to Σ+ Σ− if and only if: = 0 ∩ 0 JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 AXIOM A DIFFEOMORPHISMS DERIVED FROM ANOSOV FLOWS 43

Quadrant C +− Σ 1+ Σ 1+ Σ 0+

1/2 1/2 Σ 1/4 1+ 0 Σ 0+ -1/4

-1/2 -1/2

-3/4 Σ 0−

-5/4 Quadrant C ++ Σ 1+

Σ 0− yx y x

FIGURE 5. The surfaces Σ0 and Σ1 in the quadrants C ++ and C +−.

either y 0, z 0 and x 2/3; in this case P +(p) P −(p) (2x,0,0) • = = ≥ 0 = 0 = or y 0 z 1/2 and x 2/3; in this case P +(p) P −(p) (2x,0,1/2). • = = ≤− 0 = 0 = Σ Σ Σ Lemma 7.3 allows us to define P0 : 0 0 as P0 P0+ on 0+ and to define Σ → = P0 P0− on 0−. This map is well-defined but not injective; hence, its inverse is = 1 Σ Σ 1 1 not well-defined. In the same way, we define (P1)− : 1 1 as (P1)− (P1+)− 1 1 → = on Σ+ and (P )− (P −)− on Σ−; this map is not invertible and hence P is not 1 1 = 1 1 1 well-defined. Each of the quadrants C ++, C +−, C −+ and C −− is X -invariant. As a direct consequence, the return maps defined above preserve the quadrants. Let δ 0 such that any orbit segment of length 4δ meeting Σ is disjoint from > 0 Σ . We fix α 1 such that logα 1 log2. 1 > < 10 Σ Σ 7.4. Building f ++. Since 0+ and 1+ are disjoint global sections of the vector Σ field X restricted to C ++, every orbit in this quadrant cuts successively 0+ and Σ c c 1+. Hence every point belongs to an orbit segment [p0,p1] or [p1,p0] whose Σ Σ Σ Σ interior is disjoint from 0+ 1+ and such that p0 0+ and p1 1+. ∪ ∈ ∈c c We will first give the expression of f ++ on a segment [p0,p1] , with (p0,p1) Σ Σ Σ Σ ∩ ( 0+ 1+) , p0 0+ and p1 1+. Then f ++([p0,p1)] will be the orbit segment ∪ = ; c ∈ ∈ [P0+(p0),P1+(p1)] . Let r 4δ,s 4δ such that p X (p ,r ) and P +(p ) X (P +(p ),s). Then one > > 1 = 0 1 1 = 0 0 defines

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y x Quadrant C −−

3/4 Σ 0+ Σ 1−

1/4 0 Σ 0− 0 -1/4 Σ 1− Σ 1− 3/4 Σ 0− 1 Σ 0+

Σ 1− Quadrant C −+ y x

FIGURE 6. The surfaces Σ0 and Σ1 in the quadrants C −+ and C −−.

c c f ++ :[p ,p ] [P +(p ),P +(p )] , X (p ,t) X (P +(p ),Θ (t)), 0 1 → 0 0 1 1 0 7→ 0 0 r,s exactly as we have defined fext. c c Σ Σ Now we define f ++ on a segment [p1,p0] with (p1,p0) ( 0+ 1+) , p0 ∩ ∪ =c ; ∈ Σ+ and p Σ+: f ++([p ,p ]) will be the orbit segment [P +(p ),P +(p )] . 0 1 ∈ 1 1 0 1 1 0 0 Let r 4δ,s 4δ such that p X (p , r ) and P +(p ) X (P +(p ), s). Then > > 1 = 0 − 1 1 = 0 0 − one defines

c c f ++ :[p ,p ] [P +(p ),P +(p )] , X (p , t) X (P +(p ), Θ (t)), 1 0 → 1 1 0 0 0 − 7→ 0 0 − r,s exactly as we have deﬁned fext. One denotes by U ++ and V ++ the union of orbit segments of length 2δ centered at the points of Σ+ C ++ and Σ+ C ++, respectively. One gets: 0 ∩ 1 ∩ LEMMA 7.4. The map f ++ is well-deﬁned and is a diffeomorphism of C ++. It Σ Σ coincides with P0+ on 0+ and with P1+ on 1+. Furthermore, U ++ is an attracting region (f ++(U ++) is a subset of the interior of U ++), and f ++ contracts the X - 1 orbits segments in U ++ by the constant α− . In the same way, V ++ is a repelling 1 1 region, and (f ++)− contracts the X -orbits segments in U ++ by α− . Finally, f ++ coincides with f on {(x, y,z), x2 y2 4}. ext + ≥ JOURNAL OF MODERN DYNAMICS p VOLUME 4, NO. 1 (2010), 1–63 AXIOM A DIFFEOMORPHISMS DERIVED FROM ANOSOV FLOWS 45

Proof. The only point that is not clear “by construction” is the estimate of the radius where f ++ coincides with fext. The proof consists of noting that for every point p (x, y,z) with x2 y2 4, the segment [p ,p ]c or [p ,p ]c containing = + ≥ 0 1 1 0 p, and its image, are contained in the region (corresponding to the radius larger p 2 Σ than 3 ) where i+ coincides with Si+. For that, notice that the union of these two segments are contained in an orbit segment of length bounded by 2. Then, as X is the model vector field Xmod, the expansion of the flow on the radius is bounded by 4. 7.5. In the other quadrants. One defines in the same way the diffeomorphisms f +− : C +− C +− using the Σ Σ Σ Σ → sections 0− and 1+, f −+ : C −+ C −+ with 0+ and 1−, and f −− : C −− C −− with Σ Σ → → 0− and 1−. The X -orbit segments of length 2δ centered at the points of the corresponding sections define the corresponding attracting regions U ±± and the repelling regions V ±±. Once again, we omit the dependence on δ of the definition of f ++, f +−, f −+, and f −−. The diffeomorphisms f ++, f +−, f −+, and f −− do not coincide on the intersection of the quadrants. The next step of our construction is to glue all these diffeomorphisms together.

7.6. In a neighborhood of the periodic orbit . In this section we define a diffeomorphism f0 of a neighborhood of the periodic orbit of X . We will glue it in the next section with the diffeomorphisms we have defined in the quadrants. Note that Σ {x y 0} {(0,0,1/4),(0,0,3/4)} 0 ∩ = = = and that Σ {x y 0} {(0,0,1/2),(0,0,0)}. 1 ∩ = = = These four points will be the fixed points of f0. Henceforth, we will assume that δ 1 . < 16 One defines a Morse–Smale diffeomorphism Ψ : S1 S1 by: 0 → Ψ 1 1 1 Θ The restriction of 0 to [0, ] is t 1 , 1 (t). • 4 4 − 7→ 4 − 4 4 Ψ 1 1 1 1 Θ The restriction of 0 to [ , ] is t 1 , 1 (t). • 4 2 4 + 7→ 4 + 4 4 Ψ 1 3 3 3 Θ The restriction of 0 to [ , ] is t 1 , 1 (t) • 2 4 4 − 7→ 4 − 4 4 Ψ 3 3 3 Θ The restriction of 0 to I3 [ ,1 0] is t 1 , 1 (t) • = 4 = 4 + 7→ 4 + 4 4 Ψ 1 3 The diffeomorphism 0 has precisely 4 fixed points: two sinks ( 4 and 4 ) and 1 1 two sources (0 and 2 ). It is a linear contraction of ratio α− on the segments [ 1 δ, 1 δ] and [ 3 δ, 3 δ] and is a linear expansion of ratio α on the segments 4 − 4 + 4 − 4 + [ δ , δ ] and [ 1 δ , 1 δ ]. − α + α 2 − α 2 + α One denotes by f the diffeomorphism of R2 S1 defined as follows: 0 × 1 Ψ0(z) z (1 Ψ0(z) z) f0(p) X1 Ψ (z) z (p) (2 + − x,2− + − y,Ψ0(z)) = + 0 − = for any p (x, y,z). = JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 46 CHRISTIAN BONATTI AND NANCY GUELMAN

1 1 The diffeomorphism f0 has precisely 4 ﬁxed points: (0,0,0),(0,0, 4 )(0,0, 2 ) and (0,0, 3 ), which are saddle points. The closed X -orbit {x y 0} is a normally 4 = = hyperbolic invariant curve for f0.

7.7. Gluing f0 with f ++, f +−, f −+ and f −−. On the quadrant C ++ the surface 1 2 2 1 Σ+ contains the intersection of the horizontal disk {(x, y,z) z , x y } 0 | = 4 + < 4 with C ++, and Σ+ contains the intersection of the horizontal disk {(x, y,z) z 1 p | = 1 2 2 1 2 , x y 4 } with C ++( See Figure 5). As a consequence one gets that f ++ + < 1 1 δ coincides with f in a neighborhood, in C ++, of the segment {(0,0)} [ δ, ]. p 0 × 4 − 2 + α

DEFINITION 7.5. We define f0++ : C ++ C ++ as follows: Every point belongs to c c → an orbit segment [p ,p ] or [p ,p ] whose interior is disjoint from Σ+ Σ+ and 0 1 1 0 0 ∪ 1 such that p Σ+ and p Σ+. We distinguish 4 cases: 0 ∈ 0 1 ∈ 1 q Σ+ Σ+: • ∈ 0 ∪ 1 We define f0++(q) f ++(q). c = c q [p ,p ] with (p ,p ) (Σ+ Σ+) , p Σ+, and p Σ+: • ∈ 0 1 0 1 ∩ 0 ∪ 1 = ; 0 ∈ 0 1 ∈ 1 We define f0++(q) f ++(q). c = c q [p ,p ] with (p ,p ) (Σ+ Σ+) , p Σ+, and p Σ+, where • ∈ 1 0 1 0 ∩ 0 ∪ 1 = ; 0 ∈ 0 1 ∈ 1 2 2 1 p (x, y,z) Σ+ with x y : 1 = ∈ 1 + ≥ 100 We define f0++(q) f ++(q) c = p c q [p ,p ] with (p ,p ) (Σ+ Σ+) , p Σ+, and p Σ+, where • ∈ 1 0 1 0 ∩ 0 ∪ 1 = ; 0 ∈ 0 1 ∈ 1 Σ 2 2 1 p1 (x, y,z) 1+ with x y 100 : = ∈ + ≤1 Σ 1 1 In this case note that pp1 (x, y, 2 ) 1+ and P1+(p1) (2x, 2 y, 2 ). We have 1 = 1 ∈ 1 = p0 (x′, y′, 4 ) and P0+(p0) (2x′, 2 y′, 4 ). = = c One deduces that the image of the segment [p1,p0] under f0 and un- c der f ++ is the segment [P1+(p1),P0+(p0)] . This allows us to consider the barycentral diffeomorphism, between f0 and f ++: – consider t 0 such that f (q) X (P +(p)) 0 > 0 = t0 1 – consider t ++ 0 such that f ++(q) X (P +(p)) > = t ++ 1 – let r : x2 y2 (notice that r does not depend on q [p ,p ]c ) = + ∈ 1 0 – let t ψ(100r )t (1 ψ(100r ))t ++ = p 0 + − Then we define f ++(q) X (P +(p)). 0 = t 1

LEMMA 7.6. The map f0++ is well-deﬁned and is a diffeomorphism of the quadrant C ++ that coincides with f0 in the neighborhood of the circle {x y 0} and 2 2 2 = = coincides with f ++ in x y . Furthermore, f ++ coincides with f ++ on + ≥ 100 0 U (in particular,U is an attracting region for f ; i.e., f (U ) is included ++ ++p 0++ 0++ ++ 1 in the interior of U ++). Finally f0++ coincides with f ++ on (f ++)− (V ++) so V ++ is a repelling region. Σ Σ Proof. We know that each point x in C ++ ( 0+ 1+) belongs to a unique segment c c à ∪ of the form [p ,p ] or [p ,p ] whose interior is disjoint from Σ+ Σ+ and such 0 1 1 0 0 ∪ 1 that p Σ+ and p Σ+ and hence the map f ++ is well-deﬁned. Let 0 ∈ 0 1 ∈ 1 0 c M ++ [p ,p ] . = 1 0 {p (x,y,z) z 1 ,px2 y2 1 } 1= | =[2 + < 100 JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 AXIOM A DIFFEOMORPHISMS DERIVED FROM ANOSOV FLOWS 47

Since f ++ f ++ in C ++\M ++, it follows that the restriction of f ++ to C ++\M ++ 0 = 0 2 2 2 is a diffeomorphism. Note that if q (x′, y′,z′) satisfies x y , then = ′ + ′ ≥ 100 q C ++\M ++ and therefore f ++(q) f ++(q). ∈ 0 = p 1 2 2 1 The projection of M ++ onto {(x, y,z) z , x y } along the orbits of | = 2 + < 100 2 2 the flow, the map (x, y,z) x y , the maps ψ f , f ++ and P + and the flow → + p 0 1 X are all differentiable; therefore, the map f is a diffeomorphism restricted to p 0++ 1 1 M ++. Besides, in the case that r but close enough to , it holds that < 100 100 t ψ(100r )t (1 ψ(100r ))t ++ t ++, = 0 + − = and hence f ++ f ++. 0 = We have seen that f ++ coincides with f0 in a neighborhood of the segment 1 1 δ 1 1 {(0,0)} [ 4 δ, 2 α ] in C ++. Then in a neighborhood of {(0,0)} [ 4 δ, 4 δ] × − + 1 × − + in C ++, f0++ f ++ f0 is a contraction of factor α− . It follows that in U ++, = = 1 f0++ f ++ is a contraction of factor α− , and U ++ is an attracting region of f0++. = 1 δ 1 δ Analogously, in a neighborhood of the segment {(0,0)} [ , ] in C ++, × 2 − α 2 + α f ++ f ++ f is an expansion of factor α. Then V ++ is a repelling region for 0 = = 0 f0++.

One deﬁnes in an analogous way the diffeomorphisms f0+−, f0−+, and f0−−, verifying that they coincide with f0 in a neighborhood of the circle {x y 0}, 1 = = that they induce linear contraction of ratio α in orbit segments crossing U +−, U −+ and U −−, respectively, and that they induce linear dilations of ratio α in the 1 1 1 orbit segment crossing (f0+−)− (V +−), (f0−+)− (V −+), and (f0−−)− (V −−) respectively.

8. ONTHESIDESOFTHEQUADRANTS

The diffeomorphisms f0++ and f0+− are deﬁned on C ++ and C +− whose intersection is the half cylinder C +± {y 0,x 0}. Notice that in C +±, these = = ≥ diffeomorphisms satisfy the following properties:

f ++(x,0,z) f +−(x,0,z) f (x,0,z) in the neighborhood in C +± of the cir- • 0 = 0 = 0 cle {x y 0} that is in {0 x δ , y 0} where δ is some positive number. = = ≤ ≤ 2 = 2 f ++(x,0,z) f +−(x,0,z) for every point (x, y,z) with x 4. Furthermore • 0 = 0 ≥ f0++ and f0+− define a diffeomorphism in the neighborhood of the half affine plane {y 0,x 4} = ≥ δ f ++ f +− on the orbits segments of length 2 centered at the points of • 0 = 0 α Σ δ 1+ C +±; they map in a affine way the orbit segment of length 2 α centered ∩ Σ at p 1+ onto the orbit segment of length 2δ centered at P1+(p). ∈ 1 Note that Σ+ C +± is the horizontal half-line {y 0,x 0,z }. 1 ∩ = ≥ = 2 In other words, f0++ and f0+− coincide in the points of C +± which are outside 1 δ 1 δ of the rectangle {x [δ ,4], y 0,z [ − , ]}. ∈ 2 = ∈ 2 + α 2 − α The aim of the section is to define a diffeomorphism f +± in a neighborhood of this rectangle, with some compatibility with f ++ in C ++ and with f +− in C +−.

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1/2 Σ 1+

1/4

Σ 0+ 0

Σ 0− 1/4 −

Σ+ 1/2 1 −

Σ Σ Σ FIGURE 7. The surfaces 0−, 0+ and 1 in C +±.

This compatibility will allow us to glue it with f0++ and f0+−, preserving the properties like hyperbolicity, partial hyperbolicity, and the attracting regionsU ++ and U +−. We denote by U +± the intersection (U ++ U +−) C +± (see ﬁgure 8). We start ∪ ∩ by building a diffeomorphism f0+± of C +±, admittingU +± as an attracting region. Then f +± will be an extension of f0+± in a neighborhood and it will be deﬁned in Subsection 8.2. We will introduce analogous diffeomorphisms on neighborhoods of C −+ C −−, C −− C +− and C −+ C ++ in Subsection 8.3 and we will glue ∩ ∩ ∩ all the pieces in Subsection 8.4 getting the model diffeomorphism.

8.1. Building f0+± on C +±. As we have done several times before, we start by covering the half cylinder u C +± {x 0, y 0} with a family of X -orbit segments I (x). = ≥ = 1 REMARK 8.1. The intersection Σ+ C +± {x 0, y 0,z } is a complete sec- 1 ∩ = ≥ = = 2 tion of X on C +± and the return time on it is equal to 1. 1 In other words, for p (x,0, ) Σ+ C +± one has: P +(p) X (p,1). = 2 ∈ 1 ∩ 1 = u 1 For every x 0 we denote by I (x) the X -orbit segment joining p (x,0, 2 ) ≥ 1= to P1+(p) X (p,1). We denote by p1(x) the ﬁrst coordinate of P1+(x,0, 2 ); that is, 1= 1 P +(x,0, ) (p (x),0, ). 1 2 = 1 2 LEMMA 8.2. There is a diffeomorphism f0+± of C +± having the following properties: u u 1. f0+± maps each X -orbit segment I (x) on I (p1(x)) (in particular, f0+± preserves each X -orbit); 1 1 1 1 2. f0+± coincides with f0++ and f0+− on the set where z [ 2 α− δ, 2 α− δ], ∈ − +1 that is on the intersection of C +± with the repelling regions (f0++)− (V ++) 1 and (f0+−)− (V +−). 3. there are 0 x x 10 such that f +± coincides with f ++ and f +− on the < 0 < 1 < 0 0 0 sets {x x } and {x x }. ≤ 0 ≥ 1 JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 AXIOM A DIFFEOMORPHISMS DERIVED FROM ANOSOV FLOWS 49

Σ 1+ 1/2 δ δ

1/4 δ δ δ δ

U +± 0 δ δ

-1/4 δ δ δ δ

δ δ δ -1/2 1/3 2/3 Σ 1+

FIGURE 8. The attracting region U +± C +±. ⊂

4. For every p Σ C +±, the image of the orbit segments X (p,[ δ,δ]) is con- ∈ 0 ∩ − tained in the interior of the segment X (P0(p),[ δ,δ]) and the central deriv- 1 − ative is at most α− on this segment. (see Figure 9)

8.2. Extension of f0+± in a neighborhood of C +±. Σ Σ Σ 8.2.1. Extension of 0+ and 0− in a neighborhood of C +±. For p 1+ we denote u ∈ by I (p) the X -orbit segment joining p with P1+(p); notice that with this notation u 1 u I (x,0, 2 ) I (x) for x 0. = ≥ u For any ε 0, let W +± be the union of the X -orbit segments I (p) for p Σ+ > ε ∈ 1 ∩ { y ε}. Recall that the positive flow of X contracts uniformly the y-coordinates. | |≤ As a consequence, one gets that Wε+± is positively invariant under the flow of X . Σ Σ Recall that 0+ and 0− are smooth sections of X on C ++ and C +−, respectively such that they induce sections of X on Wε+± {y 0} and Wε+± {y 0}, respec- ∩ ≥ 2 ∩ ≤ tively. Furthermore, Σ+ and Σ− coincide on C +± for x (and fit together in a 0 0 ≥ 3 smooth surface), and each of these surfaces coincide with a horizontal disk in { x2 y2 1 }. Hence: + ≤ 3 p LEMMA 8.3. There are ε0 0 and sections Σ˜ + and Σ˜ − of X onW +± such that > 0 0 ε0 1. Σ˜ + and Σ˜ − coincide with Σ+ and Σ− on {y 0} and {y 0}, respectively; 0 0 0 0 ≥ ≤ 2 2 1 2. Each of Σ˜ + and Σ˜ − coincides with a horizontal disk on { x y } 0 0 + ≤ 3 Σ˜ Σ˜ 2 2 3. 0+ and 0− coincide on { x y 1}. p + ≥ u 4. Each of Σ˜ + and Σ˜ − cuts every segment I (p),p (x, y,z), x 0 and y ε , 0 0 p = ≥ | |≤ 0 in exactly one point.

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1/2

1/4

U +± 0

-1/4

-1/2 1/3 2/3 1 4/3 5/3

1/2

1/4

0 f +±(U +±)

-1/4

-1/2 1/3 2/3 1 4/3 5/3

FIGURE 9. The map f0+±(x, y,z) on U +±

5. The first-return maps P˜+ : Σ˜ + Σ˜ + and P˜− : Σ˜ − Σ˜ − are well-defined and 0 0 → 0 0 0 → 0 extend P0+ and P0− respectively. 6. For every 0 ε ε there is 0 ε′ ε such that < < 0 < < Σ˜ ˜ Σ˜ p + Wε+± P +(p) + W +± ∈ 0 ∩ =⇒ 0 ∈ 0 ∩ ε′ Σ˜ ˜ Σ˜ p 0− Wε+± P0−(p) 0− W +± ∈ ∩ =⇒ ∈ ∩ ε′ Σ Σ p + Wε+± P +(p) + W +± ∈ 1 ∩ =⇒ 1 ∈ 1 ∩ ε′ Proof. The sole difficulty is the last item. For that, notice that the positive flow of X contracts uniformly the y-coordinates and that the return times on these sections is strictly positive, bounded away from 0.

For 0 ε ε we denote by Σ˜ + and Σ˜ − the intersection of Σ˜ + and Σ˜ − with < ≤ 0 0,ε 0,ε 0 0 Wε+±. 8.2.2. Extension of f0+± onWε+±. One deﬁnes

τ+± : C +± (0, ) by f +±(p) X (p,τ+±(p)) 0 → +∞ 0 = 0 τ++ : C ++ (0, ) by f ++(p) X (p,τ++(p)) → +∞ 0 = τ+− : C +− (0, ) by f +−(p) X (p,τ+−(p)). → +∞ 0 = JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 AXIOM A DIFFEOMORPHISMS DERIVED FROM ANOSOV FLOWS 51

This deﬁnition implies that the map τ0+± coincides with τ++ and τ+− where f +± coincides with f0++ and f0+−. One deduces:

LEMMA 8.4. There exist x ,x , positive numbers with 0 x x 10 and a 0 1 < 0 < 1 < smooth map τ+± : W +± (0, ) with the following properties: ε0 → +∞ 1. τ+± τ0+± on C +±. = Σ 1 1 2. τ+±(p) τ++(p) for p X (q,t), q 1+ C ++, t [ α− δ,α− δ]. = = ∈ ∩ ∈ − 1 1 3. τ+±(p) τ+−(p) for p X (q,t), q Σ+ C +−, t [ α− δ,α− δ]. = = ∈ 1 ∩ ∈ − 2 2 2 2 4. τ+±(p) τ++(p) for p (x, y,z) C ++, x y x0 or x y x1. = = ∈ 2 + 2 ≤ 2 + 2 ≥ 5. τ+±(p) τ+−(p) for p (x, y,z) C +−, x y x or x y x . = = ∈ p + ≤ 0 p + ≥ 1 6. The central derivative X τ+±(p) is everywhere strictly larger than 1. · p 1 p − 7. The central derivative X τ+±(p) is at most α− 1 for p X (q,t) with q · − = ∈ Σ˜ + Σ˜ −, t [ δ,δ] 0 ∪ 0 ∈ − ˜ One deﬁnes f +± on Wε+± by

f˜+±(p) X (p,τ+±(p)). = The fact that the central derivative of τ+± is larger than 1 implies that f˜+± Σ − ˜ u is a diffeomorphism. Notice that for every p 1+ Wε+± one has f +±(I (p)) u ∈ ∩ = I (P1+(p)). ˜ Furthermore f +± coincides with f0+± on C +±. As a consequence one proves:

LEMMA 8.5. There is ε 0 such that for every p (x, y,z) Σ+ (resp. Σ− ), the > = ∈ 0,ε 0,ε image under f˜+± of the orbit segments X (p,[ δ,δ]) is contained in the interior of ˜ ˜ − the segment X (P0+(p),[ δ,δ]) (resp. X (P0−(p),[ δ,δ])). Furthermore, the central − 1− derivative of f˜+± is less than or equal than α− on this segment.

Proof. The ﬁrst property holds on C +±. It can be extended in a small neighborhood by a continuity argument using that f˜+± preserves each X -orbit. The second comes from the assumption on the central derivative X τ+±(p). · We ﬁx ε 0 as in Lemma 8.5. By omitting the dependence on ε we call 1 > 1 ˜ f +± f +± W . = | ε+±1 As a consequence of Lemma 8.3 one gets:

LEMMA 8.6. For every ε (0,ε1), there is 0 ε′ ε such that f +±(Wε+±) W +±. ∈ < < ⊂ ε′

By construction, one gets that f +± coincides with the maps f0++ f0+− in a neighborhood of the periodic orbit and for large radius:

LEMMA 8.7. There exist 0 x0 x1 10 such that for every (x, y,z) Σ+ W +±, < < < ∈ 1 ∩ ε1 one has: u If x x , then the restriction of f +± to I (x, y,z) coincides with f and • ≤ 0 0 hence with f0++ if y 0 and with f0+− if y 0; ≥ u ≤ If x x , then the restriction of f +± to I (x, y,z) coincides with f ++ if y 0 • ≥ 1 0 ≥ and with f +− if y 0. 0 ≤

JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 52 CHRISTIAN BONATTI AND NANCY GUELMAN

8.2.3. Attracting and repelling regions. For every ε 0, one denotes by V +± andU +± theunionofthe X -orbit segment > ε or length 2δ centered at Σ+ and Σ˜ + Σ˜ − , respectively. Thus, V +± V ++ V +−. 1 0,ε ∪ 0,ε = ∪ 2 2 Notice that U +± C +± U +±. Furthermore, in { x y 2}, the set U +± coin- ε ∩ = + > ε cides with U in C , and with U in C . ++ ++ +− +− p The next lemma asserts that the setsUε+± and V +± are invariant under positive and negative iterates of f +±, respectively:

LEMMA 8.8. For every 0 ε ε there is 0 ε′ ε such that < < 1 < < For every (x, y,z) U +±, the image f +±(x, y,z) belongs to the interior of • ∈ ε U +±; moreover, D f +±(x, y,z) contracts the vector ﬁeld X by a factor at most ε′ 1 1 α− , and this factor is equal to α− if x [x0,x1] (where x0,x1 are given by ∉ c Lemma 8.7). In other words, the central derivative D f +±(x, y,z) is less than 1 1 α− , and it is equal to α− if x [x ,x ]. ∉ 0 1 For every (x, y,z) W +± such that f +±(x, y,z) belongs to V +±, the point • ∈ ε (x, y,z) belongs to the interior of V +±; moreover, the central derivative, c 1 (D f +±)− (x, y,z), 1 is equal to α− 8.3. The diffeomorphisms in a neighborhood of the other sides.

8.3.1. The diffeomorphism f −± in the neighborhood of C −± C −− C −+. We = ∩ Σ proceed exactly as we have done in a neighborhood of C +±, just substituting 1+ Σ by 1−, (and changing the sign of the x): Consider the union of orbit segments u W −± I (p) ε = p (x,y,z) Σ− { y ε} = [∈ 1 ∩ | |≤ u where I (p) is the orbit segment joining a point p Σ− to its first return P −(p) ∈ 1 1 on Σ−. Then W −± is a neighborhood of the unstable separatrix C −± {x 0, y 1 ε = ≤ = 0,z S1}. ∈ Σ Σ Σ˜ Σ˜ As in the previous case we extend 0+ and 0− to 0,+ε and 0,−ε in Wε−± and we define U −± and V −±. We also define f −± : W −± W −± exactly in the same way ε ε → ε as f +±. We get 0 ε ε and 0 x x 10 such that Lemmas 8.3, 8.5, 8.6, 8.7, < 1 < 0 < 0 < 1 < and 8.8 hold in this new context.

8.3.2. The diffeomorphisms f ±+ and f ±− in neighborhoods of C ++ C −+ and ∩ C +− C −−. As we already noticed, it is more convenient to consider the inverse ∩ of the return maps in the neighborhood of the stable manifold {x 0} because = the flow expands the x coordinates: as a consequence, the set { x ε} is not | |≤ invariant for the positive times of the flow but is invariant for negative times. For this reason we define:

s W ±− I (x, y,z) ε = (x,y,z) Σ− { x ε} ∈[0 ∩ | |≤ JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 AXIOM A DIFFEOMORPHISMS DERIVED FROM ANOSOV FLOWS 53

s Σ where I (x, y,z) is the orbit segment joining a point (x, y,z) 0− to its ﬁrst nega- 1 Σ ∈ tive return (P0−)− (x, y,z) on 0−. Analogously we deﬁne

s W ±+ I (x, y,z) ε = (x,y,z) Σ+ { x ε} ∈[0 ∩ | |≤ s Σ where I (x, y,z) is the orbit segment joining a point (x, y,z) 0+ to its ﬁrst nega- 1 Σ ∈ tive return (P0+)− (x, y,z) on 0+. Analogous to the construction of f +± and f −±, we build diffeomorphisms 1 1 1 (f ±+)− : W ±+ W ±+ and (f ±−)− : W ±− W ±− that coincide with (f )− for ε → ε ε → ε 0 y y , and with the inverse of f ++, f +−, or f −+, f −− (according to the corre- | |≤ 0 0 0 0 0 sponding quadrant), for y y , for some 0 y y 10. | |≥ 1 < 0 < 1 < As in the previous cases, we deﬁne U ±+, Vε±+, U ±− and Vε±−. The set U ±+ is strictly invariant under f ±+, and U ±− is strictly invariant under f ±−, and their 1 central derivatives are equal to α− . Moreover Vε±+ is an attracting region for 1 1 (f ±+)− , and Vε±− is an attracting region for (f ±−)− , for every small ε and the 1 1 1 central derivative of (f ±+)− and of (f ±−)− are at most α− , and it is equal to 1 α− if y [y , y ]. | | ∉ 0 1 In that way, we get 0 ε ε and 0 y y 10 such that statements, < 1 < 0 < 0 < 1 < equivalent to Lemmas 8.3, 8.5, 8.6, 8.7, and 8.8, hold for the inverse diffeomorphisms, exchanging the role of the coordinates x and y.

8.4. Gluing all the pieces of the puzzle. We will now glue the diffeomorphisms f0++, f0+−, f0−+, and f0−− we have deﬁned in the respective quadrants with the diffeomorphisms f +±, f −±, f ±+, and f ±− in order to get a diffeomorphism of R2 S1 that will be our local model of an Axiom A diffeomorphism in a neighbor- × hood of the boundary component of a Birkhoff section. Let us now make some easy observations that will help us in this construction. 1. All these diffeomorphisms preserve every leaf of the 1-dimensional foliation generated by the vector ﬁeld X . More precisely, each of them is of the form (x, y,z) X (x, y,z), where t is a strictly positive number depending 7→ t smoothly on (x, y,z). 2. Furthermore, for f ++,..., f −−, f0, f +±,..., f ±−, the orbit segment joining a Σ˜ s point to its image is larger than the smallest orbit segment joining i,ε to Σj , i j {0,1}, and s { , } and smaller than two times the largest time 6= ∈Σ Σ ∈ + − 1 5 return on i+ and on i−. For ε 0 these time distances are 4 and 2 4 respec- = 1 tively. For small ε, these time distances are larger than 5 and smaller that 3, respectively. Hence all these diffeomorphisms are of the form p Xt (p) 1 7→ with t [ ,3]. The diffeomorphisms f ++,..., f −− are obtained as barycen- ∈ 5 0 0 ters along the orbits of the diffeomorphism f0 with f ++,..., f −−. One deduces that the same estimates hold for these diffeomorphisms. 3. The neighborhoods Wε+± and Wε−± of the unstable separatrices are invariant under these diffeomorphisms. More precisely, W +± C ++ is invariant ε ∩ JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 54 CHRISTIAN BONATTI AND NANCY GUELMAN

under f0++ and f +±. Analogously, the intersections of each of these neighborhoods Wε+± and Wε−± with each quadrant are invariant under the corresponding diffeomorphisms. u 4. Recall that Wε+± is union of orbit segments I (x, y,z) with extremal points Σ u u (x1, y1,z1) 1+. The images f0++(I (x, y,z)) and f +±(I (x, y,z)) are both ∈ u the same orbit segment I (P1+(x1, y1,z1)). Moreover, f0++ and f +± coin- u 1 cide on the two extremal subsegments of I (x, y,z) of length α− δ with the afﬁne dilation of ratio α. The same happens at the intersection of any Wε+± and Wε−± with each quadrant. 5. The neighborhoods Wε±+ and Wε±− of the stable separatrices are invariant under the inverse of these diffeomorphisms. More precisely, Wε±+ C ++ 1 1 ∩ is invariant under (f0++)− and (f +±)− . Analogously, the intersections of each of these neighborhoods Wε±+ and Wε±− with each quadrant are invariant under the inverse of the corresponding diffeomorphisms. s Σ 6. Wε±+ is the union of orbit segment I (x, y,z) with extremal points in 0+. 1 s 1 s The images (f0++)− (I (x, y,z)) and (f +±)− (I (x, y,z)) are both the same s 1 1 1 orbit segment I (P0− (x, y,z)). Moreover, (f0++)− and (f +±)− coincide on s 1 the two extremal subsegments of I (x, y,z) of length α− δ with the afﬁne dilation of ratio α. The same happens at the intersection of any Wε±+ and Wε±− with each quadrant. 7. There is a neighborhood of the periodic orbit such that all the diffeomor-

phisms f0++, f0+−, f0−+, f0−− f +±, f −±, f ±+ and f ±− coincide with f0 (and 1 their inverses coincide with f0− ). Hence (by shrinking ε if necessary) we can assume that for every (x, y,z) ∈ Wε+± Wε±+ then all these diffeomorphisms and their inverses coincide ∩ 1 u s with f0 and f0− , respectively, on I (x1, y1,z1) and on I (x2, y2,z2), where u s I (x1, y1,z1) and I (x2, y2,z2) are the orbit segments containing (x, y,z). The same happens for the other intersections of W +± W −± with W ±+ ε ∪ ε ε ∪ Wε±− in the other quadrants. We are now ready for gluing the pieces of the puzzle.

DEFINITION 8.9. We denote by fm the diffeomorphism deﬁned as follows: If (x, y,z) belongs to the complement of • 1 1 W +± W −± (f ±+)− (W ±+) (f ±−)− (W ±−), ε ∪ ε ∪ ε ∪ ε

then fm(x, y,z) is f0++(x, y,z), f0+−(x, y,z), f0−+(x, y,z) or f0−−(x, y,z) according to the quadrant containing (x, y,z). If (x, y,z) W +± C ++, let (x , y ,z ) Σ+ and t ,t such that: • ∈ ε ∩ 1 1 1 ∈ 1 1 2 – (x, y,z) I u(x , y ,z ) ∈ 1 1 1 – f +±(x, y,z) X (P +(x , y ,z )). = t1 1 1 1 1 – f0++(x, y,z) Xt2 (P1+(x1, y1,z1)). = y1 y1 Then f (x, y,z) X (P +(x , y ,z ) where r ψ( )t (1 ψ( ))t . We m = r 1 1 1 1 = ε 1 + − ε 2 deﬁne f exactly in the same way on the intersection of W +± W −± with m ε ∪ ε each of the quadrants;

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We have already defined f in the complement of • m 1 1 (f ±+)− (W ±+) (f ±−)− (W ±−). ε ∪ ε 1 1 We will now define fm− : Wε±+ C ++ (f ±+)− (Wε±+) C ++. ∩ →Σ ∩ If (x, y,z) Wε±+ C ++, let (x1, y1,z1) 0+ and t1,t2 such that: ∈ s ∩ ∈ – (x, y,z) I (x1, y1,z1) 1∈ 1 – (f ±+)− (x, y,z) Xt1 ((P0+)− (x1, y1,z1)). 1 = 1 – (f0++)− (x, y,z) Xt2 ((P0+)− (x1, y1,z1)). 1 = 1 x1 x1 Then (fm)− (x, y,z) Xr ((P0+)− (x1, y1,z1), where r ψ( ε )t1 (1 ψ( ε ))t2. = =1 + − Considering the inverse map, this defines f : (f ±+)− (W ±+) C ++ W ±+ m ε ∩ → ε ∩ C ++. We define fm exactly in the same way on the intersection of 1 1 (f ±+)− (W ±+) (f ±−)− (W ±−) ε ∪ ε with each of the quadrants.

LEMMA 8.10. The map f deﬁned above is well-deﬁned on R2 S1 and it is a m × diffeomorphism of R2 S1. ×

Proof. We have that the maps f +±, f −±, f ±+, f ±−, f0++, f0+−, f0−+ and f0−− are dif- u feomorphisms that map central arcs of the form I (x1, y1,z1) with (x1, y1,z1) Σ1 u s ∈ to I (P1(x1, y1,z1)) and their inverses map central arcs of the form I (x0, y0,z0) Σ s 1 with (x0, y0,z0) 0 to I (P0− (x0, y0,z0)). Note that the coefﬁcient of the barycen- ∈ u s ter is constant on each segment I (x1, y1,z1) and I (x1, y1,z1).

8.5. Proof of Lemma 6.3.

PROPOSITION 8.11. The diffeomorphism fm has all the properties announced in Lemma 6.3

Proof. Most of these properties have already been proven. The times estimates t [ 1 ,3] comes from the fact that f is obtained as a barycenter along the orbits ∈ 5 m of diffeomorphisms satisfying the same time estimates (see item 2 of the list of properties in Section 8.4). For every 0 ε′ ε we deﬁne < < Uε U +± U −± U ++ U +− U −+ U −− ′ = ε′ ∪ ε′ ∪ ∪ ∪ ∪ and

Vε V ±+ V ±− V ++ V +− V −+ V −− ′ = ε′ ∪ ε′ ∪ ∪ ∪ ∪ Each of the sets U +±,U −±,U ++,U +−,U −+,U −− is invariant under its respec- ε′ ε′ tive diffeomorphism. Furthermore, we have seen that in { x2 y2 2}, the set + > U +± coincides with U ++ in C ++, and with U +− in C +− (and the same happens ε′ p for U −±, V ±+ and V ±−). ε′ ε′ ε′

CLAIM 8.12. For ε′ (0,ε) small enough, fm(Uε′ ) is contained in the interior of 1 ∈ Uε′ and fm− (Vε′ ) is contained in the interior of Vε′ .

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Proof. Consider an X -orbit segment σ of length 2δ centered at a point p of Σ0. On σ, fm is a barycenter of two diffeomorphisms, each of them mapping σ into the interior of the X -orbit segment of length 2δ centered at P0(p). Hence, the same holds for the image under fm of σ. Consider now an X -orbit segment σ of length 2δ centered at a point p of Σ˜ + 0,ε′ or Σ˜ − . Then for small ε′, fm coincides with f +± or f −± (according to the quad- 0,ε′ rants). Hence, fm(σ) is contained in the interior of the orbit segment of length ˜ ˜ 2δ centered at P0+(p) or P0−(p) which is contained in the interior of Uε′ . 1 The control of the central derivative of fm on Uε′ and of fm− on Vε′ comes from the fact that fm is obtained as a barycenter (with coefﬁcient constant along the orbit segments) of diffeomorphisms satisfying the announced estimates. c Notice that the set of points of Uε′ where the central derivative D fm is differ- 1 ent of α− is contained in Uε′ { x [x0,x1]} { y ε}, so in this compact set ∩ | | ∈ ∩ | |≤ c there is a constant β 0 that is a lower bound for the central derivatives D (fm). > N It follows that there is N such that for any point (x, y,z) Uε′ there are at most N ∈ i ∈ N numbers i such that fm(x, y,z)) Uε′ { x [x0,x1] { y ε}, then there ∈ ∈ c ∩ i| | ∈ ∩ 1| |≤ are at most N numbers i N such that D f (f (x, y,z)) α− . It follows that for ∈ m m 6= all α′ α, there is N 0 such that for any n N for any point (x, y,z) U one > > ≥ ∈ ε′ has c n n n D f (x, y,z) (α′− ,α− ]. m ∈

The proofs for Vε′ are analogous.

The rest of the items of Lemma 6.3 hold by construction of Uε′ and fm. We rename Umod to Uε′ and fmod to fm to get the notation of Lemma 6.3.

9. DYNAMICAL PROPERTIES OF THE DIFFEOMORPHISMS fr

9.1. The dynamics of fr and the pseudo-Anosov map on the Birkhoff section. The aim of this section is to describe dynamical properties of fr (built in Sec- tion 6.3) that are not direct consequences of Lemma 6.3: these properties are deeply related to the precise construction of the model. More precisely: Let N be the closed surface obtained from the Birkhoff section B0 by identi- fying each boundary component with a point. We denote by Sing N the ﬁnite ⊂ set of points pγ correspond to the boundary components γ of B0. Fried showed in [11] that the ﬁrst-return map PB0 induces on N a pseudo-Anosov homeomorphisms P : N N, whose singular points are contained in Sing. We will show: N → PROPOSITION 9.1.

1. For every large r , the Axiom A diffeomorphism fr has exactly two basic pieces, one of them is an attractor and the other is a repeller; more precisely, with the notation of Section 6.3, the maximal invariant sets Ar and Rr of fr in Ur and Vr , respectively, are a connected transitive (indeed mixing) attractor and repeller, respectively; 2. There is a continuous surjective projection π : A N that induces a semi- Ar r → conjugacy between f and P . Furthermore r |Ar N JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 AXIOM A DIFFEOMORPHISMS DERIVED FROM ANOSOV FLOWS 57

1 The preimage π− (p) is a single point if p does not belong to the unstable • Ar manifold of a point in Sing; 1 The preimage π− (p) consists of exactly two points in the same central • Ar leaf if p belongs to any unstable separatrix of a point in Sing; and 1 The preimage π− (p) consists of 2n periodic points if p pγ Sing, • Ar = ∈ where the linking number of γ is n; in this case, the preimage of each ± unstable separatrix of pγ consist of the union of two unstable separatri- 1 ces corresponding to two successive periodic points in π− (p ). Ar γ

REMARK 9.2. Item 2 of Proposition 9.1 means that Ar is obtained from the pseudo Anosov map PN by “opening” the unstable separatrices of the periodic point in Sing. That is, the attractor Ar is obtained as a “derived from pseudo-Anosov map”, which is the analog of the derived from Anosov construction in [31].

REMARK 9.3. Item 1 of Proposition 9.1 and Proposition 6.9 end the proof of The- orem 1.3.

We start the proof by building a surface BΣ, with boundary and corners, obtained by cutting the Birkhoff section B0 along the local unstable manifolds of the orbits γ in ∂B.

9.2. The surface BΣ. Let us denote by BΣ the surface with boundary and corners 1 that coincides with B outside of O and with (h Γ )− (Σ ) on O , where 0 γ γ r ◦ γ 0,n γ n is the linking number of γ. We denote by PΣ : BΣ BΣ the map that coincides S → with P outside O and that is induced (via conjugacy by h Γ ) by the map P B0 γ r ◦ γ 0 on each Oγ (recall that P0 is the map built in Section 7.3, which coincides with Σ Σ P0+ on 0+ and with P0− on 0−). Let UΣ be the union of the orbit segments of length 2δ centered at the points of BΣ.

LEMMA 9.4. A UΣ. r ⊂ Proof. Let us denote by BΣ,ε the branched surface with boundary and corners 1 that coincides with B0 outside of γ Oγ and with (hr Γγ)− (Σ0,n,ε) on Oγ, where ◦ 2 n is the linking number of γ and Σ is a lift of Σ+ Σ− to R R/ n Z. Let S 0,n,ε 0,ε ∪ 0,ε × | | UΣ,ε be the union of the orbit segments of length 2δ centered at the points of BΣ . Then for all ε there is ε′ ε such that P : eΣ e Σ and therefore ,ε < 0,ε 0,n,ε → 0,n,ε′ one has that f (UΣ ) UΣ . One deduces that the maximal invariant set of f in r ,ε ⊂ ,ε′ r UΣ is contained in UΣ UΣ. ,ε ,0 = LEMMA 9.5. Let p,q BΣ. Assume that there is t [0,2δ] such that X (p) q. ∈ ∈ t = Then either p q or p and q belong to ∂BΣ. = Proof. The statement is true by construction if p or q are not in Oγ for some γ (δ is much smaller than the return times on B0). Hence, it is enough to verify the same statement for the lift Σ0,n of the model, then it is enough to prove the statement for the model Σ on R2 S1. The statement is true if both p and q 0 × belong to Σ+ or if both belong to Σ−. So we can assume p Σ+ and q Σ− (the 0 0 ∈ 0 ∈ 0 case p Σ− and q Σ+ is analogous). The fact that p and q are on the same ∈ 0 ∈ 0 JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 58 CHRISTIAN BONATTI AND NANCY GUELMAN orbit segment of X implies that p and q belong to the plane y 0 which mod = is the unstable manifold of the periodic orbit of Xmod. If p or q belong to the Σ Σ intersection 0+ 0−, then p q because, by construction, the ﬁrst return time ∩ Σ = Σ of Xmod of such a point in 0 is much larger than 2δ. Hence, p 0+ {(x, y,z), y 2 2 2 2 ∈ ∩ = 0,x [ , ]} and q Σ− {(x, y,z), y 0,x [ , ]}; that is, p and q belong to ∈ − 3 3 ∈ 0 ∩ = ∈ − 3 3 the boundary of Σ0. Σ 9.3. Projections of the surface BΣ on B0 and on N. Notice that the surfaces 0+ Σ and 0− have been obtained by pushing the points of S0+ and S0− along the orbits Σ of Xmod. The inverse of this construction induces diffeomorphisms π0+ : 0+ 1 1 1 1 à {(0,0)} S S+ {(0,0)} S and π− : Σ− {(0,0)} S S− {(0,0)} S . One × → 0 à × 0 0 à × → 0 à × can easily verify that π+ π− on Σ+ Σ−, inducing a diffeomorphism π : Σ 0 = 0 0 ∩ 0 0 0 à {(0,0)} S1 S {(0,0)} S1. Furthermore, π is the identity map on Σ S , × → 0 à × 0 0 ∩ 0 that is, on the set {(x, y,z) Σ x2 y2 2/3}. ∈ 0 | + ≥ Let π˜ Σ : BΣ ∂B B ∂B be the map that is the identity map outside of the à 0 → 0 à 0 p Oγ and is induced by the map π0 built above in each of the Oγ. Let us state some properties of π˜ Σ: 1 1 π˜ Σ(p) X (p) for some t [ , ], for every p BΣ ∂B . • = t ∈ − 4 4 ∈ à 0 if π˜ Σ(p) π˜ Σ(q) for p q then p and q belong to the boundary of BΣ. • = 6= Notice that π˜ Σ may be seen as a continuous projection from BΣ ∂B on N à 0 à Sing B ∂S . One easily veriﬁes that π˜ Σ extends in a unique way in a projection = 0à 0 πΣ : BΣ N. → REMARK 9.6. πΣ is injective on BΣ ∂BΣ. • à P πΣ πΣ PΣ; that is, πΣ induces a semi conjugacy between PΣ and the • N ◦ = ◦ pseudo-Anosov map PN

COROLLARY 9.7. Consider a segment of X -orbit I included in UΣ. Then I BΣ ∩ consists of at most 2 point p, q (p q p,q ∂BΣ) whose images by πΣ are equal 6= ⇒ ∈ πΣ(p) πΣ(q) = Let π : UΣ N be the map defined as follows: for every q UΣ there is a UΣ → ∈ point p BΣ such that q belongs to the orbit segment of length 2δ centered at ∈ p. One defines π (q) πΣ(p). The corollary above implies that this point is UΣ = well-defined. Furthermore:

COROLLARY 9.8. The map π : UΣ N is continuous and induces a semi conju- UΣ → gacy between the restriction of fr to UΣ and the pseudo Anosov homeomorphism PN . Proof. The semiconjugacy property can be easily deduced from the semiconjugacy property of πΣ and the fact that if q belongs to the orbit segment of length 2δ centered at p BΣ, then fr (q) is contained in the orbit segment of length 1 ∈ 2α− δ centered at PΣ(p). 9.4. Proof of Proposition 9.1.

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Π 1(p) component of W u(γ) B Σ− ∩ 0

u W (γ) BΣ ∩

2/3 1/2 0 ΠΣ 0 1/2 2/3 2/3

BΣ ∂B ∩ 0 ∂B0 Π 3 Σ Σ S0 the surface 0,3

FIGURE 10. The surface N in a neighborhood of a 6-prong singularity, the surface S (B in a neighborhood of γ with n(γ,B) 0,3 0 = 3 ) and the surface Σ0,3.

9.4.1. Transitivity and semiconjugacy with the pseudo-Anosov map. The diffeomorphism fr satisﬁes Axiom A and strong transversality. Furthermore, the compact set UΣ,ε is an attracting region (its image is included in its interior). So it contains a basic piece of fr , which is a transitive attractor Λ; it is contained in the maximal invariant set Ar and is hence contained in UΣ. The attractor Λ cannot be a periodic point because fr is partially hyperbolic and the strong unstable bundle has dimension equal to 1. Recall that by Claim 4 of the proof of Proposition 6.9, the dimension of the stable direction of every point in Ar is 2. So the unstable manifold of every point in Λ has dimension 1. Consider any point p Λ. As Λ is an attractor, the unstable manifold W u(p) is ∈ contained in Λ and hence in UΣ. Consider a point p Λ that does not belong to the unstable manifold of a ∈ u periodic point q ∂BΣ, and let us denote by Λ Λ the closure of W (p). Recall ∈ 0 ⊂ that every basic piece of an Axiom A diffeomorphism splits into a disjoint union of compact sets that are cyclically permuted by the diffeomorphism, and such that the return map is mixing (see [31]); the compact sets of this decomposition

JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 60 CHRISTIAN BONATTI AND NANCY GUELMAN will be called the mixing components of the basic piece. Furthermore, these mixing components of a basic piece are built as follows: consider a point in the basic piece, the corresponding mixing component is the closure of the transverse intersection between its invariant (stable and unstable) manifolds. Hence, Λ0 is the mixing component of Λ that contains p; let i denote the period of Λ0. u Using Corollary 9.8, one deduces that the projection πUΣ (W (p)) is a (regular) leaf of the unstable foliation of PN . As PN is a pseudo-Anosov map, every unstable leaf is dense in N. As a consequence, one gets that π (Λ ) N. UΣ 0 = This implies that for every p BΣ ∂BΣ, the orbit segment of length 2δ cen- ∈ à tered at p contains at least a point in Λ0. By compactness of Λ0, it follows that for every p BΣ, the orbit segment of length 2δ centered at p contains at least a ∈ point in Λ0. j Assume now that K A is a compact set that is invariant under f for some ⊂ r r j 0 with K Λ , and let q be a point of K . As K A UΣ, there is q BΣ > 6= 0 ⊂ r ⊂ 0 ∈ such that q belongs to the orbit segment I of length 2δ centered at q0, and this n j segment contains a point p Λ . Now the iterates f (I), n N, contain the ∈ 0 r ∈ n j n j Λ n j points fr (q) K and fr (p) 0; furthermore, the length ℓ(fr )(I) is bounded n j ∈ ∈ by 2δα− and hence tends to 0. This implies that K Λ . Therefore, UΣ ∩ 0 6= ; does not contain any mixing component of a basic piece other than Λ0, this furthermore implies that Λ Λ , and it is a connected mixing attractor. = 0 This ends the proof of item (1) of Proposition 9.1 (and therefore of Theo- rem 1.3): the compact set Ar is the unique (connected and mixing) attractor of fr . 9.4.2. Injectivity defect of the semiconjugacy. Let us now prove item (2). We have already seen that the restriction of πUΣ to Ar is a continuous surjection on N inducing a semiconjugacy between fr and PN . Let us prove that the restriction of πUΣ to Ar is injective on the complement of the unstable manifolds of the periodic orbits in A ∂B . r ∩ 0 CLAIM 9.9. Let p,q BΣ be two different points, and let I UΣ and J UΣ be ∈ ⊂ ⊂ the maximal orbit segment containing p and q, respectively (recall that these segments have length bounded by 4δ ). Assume that fr (I) and fr (J) are contained in the same orbit segment, L UΣ. Then p ∂BΣ and q ∂BΣ. ⊂ ∈ ∈ Proof. By construction, fr (I) and fr (J) are contained in the orbit segment in UΣ through the points PΣ(p) and PΣ(q), respectively. Now Lemma 9.5 implies that either PΣ(p) PΣ(q) or these points belong to the boundary of BΣ. In both cases, = p and q belong to ∂BΣ: the map PΣ is injective outside of ∂BΣ and the preimage of ∂BΣ is contained in ∂BΣ.

CLAIM 9.10. For every connected component γ ∂B the intersection W u(γ) A ⊂ 0 ∩ r is the union of the (1-dimensional) unstable manifolds W u(p) for p Per(f ) ∈ ∩ A γ. r ∩ Proof. If q W u(γ) A , then the α-limit of q is equal to one of the periodic ∈ ∩ r orbit contained in γ (because γ is a normally hyperbolic invariant circle and the

JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 AXIOM A DIFFEOMORPHISMS DERIVED FROM ANOSOV FLOWS 61 restriction of fr to γ is Morse–Smale). As the α-limit is contained in Ar we proved the claim.

Consider now a point p A that does not belong to the unstable manifold of ∈ r a point in Per(f ) ∂B A . Let q BΣ such that p belongs to the maximal orbit r ∩ 0 ∩ r ∈ segment I UΣ containing q. We will show the following: ⊂ CLAIM 9.11. I A {p}. ∩ r = Proof. Consider a point p1 Ar I. ∈ ∩ u According to Claim 9.10, the segment I is disjoint from W (γ) for all γ ∂B0. 1 ⊂ So Claim 9.9 implies that fr− (I) contains a unique point q1 in BΣ. One de- 1 1 duces by Claim 9.9 that fr− (p) and fr− (p1) belong to the the maximal orbit segment in UΣ containing q1. Hence we can iterate the process, building a sequence of points qn BΣ such that the maximal orbit segment, In in UΣ contain- n∈ n ing qn contains fr− (p) and fr− (p1). Let Jn In be the orbit segment joining n n ⊂ fr− (p) to fr− (p1). As Jn is contained in UΣ the length of these segments satis- 1 1 n ﬁes: ℓ(Jn 1) α− ℓ(Jn) α− 4δ. So ℓ(J0) 4δα− for every n; that is, p p1. − ≤ ≤ ≤ = This ends the proof of the claim.

This proves that the projection πUΣ is injective on the complement of the unstable manifolds of the periodic orbits in A ∂B . For ending the proof of the r ∩ 0 Σ proposition it remains to consider the restriction of πUΣ to the intersection of U u with the unstable manifold W (γ), where γ is a component of ∂B0.

9.4.3. Dynamics of the repeller. Note that in an analogous way, it can be proved that there is a continuous surjective projection π : R N that induces a semi- Rr r → conjugacy between f and P . Furthermore, r |Rr N 1 The preimage π− (p) is a single point if p does not belong to the stable • Rr manifold of a point in Sing; 1 The preimage π− (p) consists of exactly two points in the same central leaf • Rr if p belongs to one stable separatrix of a point in Sing; and 1 The preimage π− (p) consists of 2n periodic points if p pγ Sing, where • Rr = ∈ the linking number of γ is n; in that case, the preimage of each stable sep- ± aratrix of pγ consist of the union of two stable separatrices corresponding 1 to two successive periodic points in π− (p ). Rr γ Acknowledgments. We thank the referees for their careful reading and sugges- tions, which helped us to improve the presentation of our result. We thank both the I.M.B and I.M.E.R.L. for their warm hospitality.

REFERENCES

[1] T. Barbot, Plane afﬁne geometry and Anosov ﬂows, Ann. Sci. École Norm. Sup. (4), 34 (2001), 871–889. [2] C. Bonatti and L. Diaz, Persistent nonhypebolic transitive diffeomorphisms, Annals of Mathe- matics (2), 143 (1996), 357–396.

JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 62 CHRISTIAN BONATTI AND NANCY GUELMAN

[3] C. Bonatti, L. Diaz and M. Viana, “Dynamics Beyond Uniform Hyperbolicity," A global geometric and probabilistic perspective, Encyclopaedia of Mathematical Sciences, 102, Mathe- matical Physics, III, Springer-Verlag, Berlin, 2005. [4] C. Bonatti and N. Guelman, Transitive Anosov flows and Axiom A diffeomorphisms, to appear at Erg. Th & Dyn. Sys.. [5] C. Bonatti and R. Langevin, Difféomorphismes de Smale des surfaces, Astérisque, No. 250, (1998), viii+235 pp.. [6] C. Bonatti and A. Wilkinson, Transitive partially hyperbolic diffeomorphisms on 3-manifolds, Topology, 44 (2005), 475–508. [7] M. Brunella, Surfaces of section for expansive flows on three-manifolds, J. Math. Soc. Japan, 47 (1995, 491–501. [8] C. Conley, “Isolated Invariant Sets and the Morse Index," CBMS Regional Conference Series in Mathematics, 38, American Mathematical Society, Providence, R.I., iii+89 pp. 1978. [9] S. Fenley, The structure of branching in Anosov flows of 3-manifolds, Comment. Math. Helv., 73 (1998), 259–297. [10] J. Franks, Anosov diffeomorphisms, 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) pp. 61–93 Amer. Math. Soc., Providence, RI. [11] D. Fried, Transitive Anosov flows and pseudo-Anosov maps, Topology, 22 (1983), 299–303. [12] J. Franks and R. Williams, “Anomalous Anosov Flows," Global Theory and Dynamical Sys- tems, (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), pp. 158–174, Lecture Notes in Math., 819, Springer, Berlin, 1980. [13] E. Ghys, Codimension-one Anosov flows and suspensions, Dynamical systems, Valparaiso 1986, 59–72, Lecture Notes in Math., 1331, Springer, Berlin, 1988. [14] V. Grines, On the topological classification of structurally stable diffeomorphisms of surfaces with one-dimensional attractors and repellers, (Russian) Mat. Sb., 188 (1997), 57–94; translation in Sb. Math., 188 (1997), 537–569 . [15] V. Grines and E. Zhuzhoma, On structurally stable diffeomorphisms with codimension-one expanding attractors, Trans. Amer. Math. Soc., 357 (2005), 617–667 (electronic). [16] N. Guelman, On the approximation of time-one map of Anosov Flows by Axiom A diffeomorphisms, Bull. Braz. Math. Soc., New Series, 33 (2002), 75–97. [17] M. Grayson, C. Pugh and M. Shub Stably ergodic diffeomorphisms, Annals of Mathematics (2), 140 (1994), 295–329. [18] A. Hammerlindl, “Leaf Conjugacies on the Torus," PhD Thesis (www.math.utoronto.ca/andy/thesis.pdf). [19] Sh. Hayashi, Connecting invariant manifolds and the solution of the C 1 stability and Ω- stability conjectures for flows, Ann. of Math. (2), 145 (1997), 81–137. [20] M. Hirsch, C. Pugh and M. Shub, “Invariant Manifolds," Lecture Notes in Mathematics, vol. 583, Springer-Verlag, Berlin-New York, 1977. [21] A. Katok and B. Hasselblat, “Introduction to the Modern Theory of Dynamical Systems," With a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. [22] R. Mañé, A proof of the C 1 stability conjecture, Inst. Hautes Études Sci. Publ. Math. No. 66, (1988), 161–210. [23] S. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761– 770. [24] J. Palis, Vector fields generate few diffeomorphisms, Bull. Amer. Math. Soc., 80 (1974), 503–505. [25] J. Palis and C. C. Pugh, Fifty problems in dynamical systems, Dynamical systems—Warwick 1974 (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E. C. Zeeman on his fiftieth birthday), 345–353, Lecture Notes in Math., Vol. 468, Springer, Berlin, 1975. [26] M. Peixoto, On the classification of flows on 2-manifolds, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), 389–419, Academic Press, New York, 1973.

JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 AXIOM A DIFFEOMORPHISMS DERIVED FROM ANOSOV FLOWS 63

[27] J. W. Robbin, A structural stability theorem, Ann. of Math. (2), 94 (1971), 447–493. [28] C. Robinson, Structural stability of C 1 diffeomorphisms, J. Differential Equations, 22 (1976), 28–73. [29] F.Rodriguez Hertz, M. Rodriguez Hertz, R. Ures, A non-dynamically coherent example in T 3, in preparation. [30] S. Simić, Codimension-one Anosov flows and a conjecture of Verjovsky, Ergodic Theory Dy- nam. Systems, 17 (1997), 1211–1231. [31] S. Smale, Differentiable dynamical systems, Bull. A.M.S., 73 (1967), 747–817. [32] A. Verjovsky, Codimension-one Anosov flows, Bol. Soc. Mat. Mexicana, 19 (1974), 49–77.

CHRISTIAN BONATTI : Université de Bourgogne, Institut de Mathé- matiques de Bourgogne, UMR 5584 du CNRS, BP 47 870, 21078, Dijon Cedex, France

NANCY GUELMAN : IMERL, Facultad de Ingeniería, Universidad de La República, C.C. 30, Montevideo, Uruguay

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