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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 ex- ample, 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 dif- feomorphisms 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 hyperbolic- ity. A compact invariant set K for a diffeomorphism f is hyperbolic if the tan- gent 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. INTHEPUBLICDOMAINAFTER 2038 1 ©2010AIMSCIENCES 2 CHRISTIAN BONATTI AND NANCY GUELMAN 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 ex- panded 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 diffeo- morphisms f X ( ,t), t R cannot be hyperbolic. t = · ∈ 1.2. Anosov diffeomorphisms and vector fields. A vector field X (a diffeomor- phism 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 diffeomor- phism 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 dif- feomorphisms 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 neg- ative curvature and suspension of T 2 hyperbolic linear automorphisms. JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 1 (2010), 1–63 AXIOM A DIFFEOMORPHISMS DERIVED FROM ANOSOV FLOWS 3 Yet, there are exotic examples [5], so classification remains an open prob- lem (see [1, 9] for works in this direction). The corresponding case for dif- feomorphisms 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 hy- perbolic dynamical systems. Such a classification has been achieved in dimen- sion 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 transversal- ity 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 codi- mension 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-lami- nation 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.
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