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ON the HYPERBOLICITY of HOMOCLINIC CLASSES Christian DISCRETE AND CONTINUOUS doi:10.3934/dcds.2009.25.1143 DYNAMICAL SYSTEMS Volume 25, Number 4, December 2009 pp. 1143–1162 ON THE HYPERBOLICITY OF HOMOCLINIC CLASSES Christian Bonatti Institut de Math´ematiques de Bourgogne Universit´ede Bourgogne, Dijon 21004, France Shaobo Gan and Dawei Yang School of Mathematical Sciences Peking University, Beijing 100871, China (Communicated by Sebastian van Strien) Abstract. We give a sufficient criterion for the hyperbolicity of a homoclinic class. More precisely, if the homoclinic class H(p) admits a partially hyper- s s bolic splitting TH(p)M = E ⊕< F , where E is uniformly contracting and dim Es = ind(p), and all periodic points homoclinically related with p are uni- formly Eu-expanding at the period, then H(p) is hyperbolic. We also give some consequences of this result. 1. Introduction. 1.1. Hyperbolicity and periodic orbits. The hyperbolicity is one of the funda- mental tools for studying dynamical systems, in particular in the last 50 years with the development of Smale’s hyperbolic theory. A fixed point p of a diffeomorphism f is hyperbolic, if the norm of every eigenvalue of the derivative Dpf is not equal to 1. This is equivalent to say that there are C > 0, s u 0 <λ< 1 and an invariant splitting TpM = E ⊕ E , such that for any n ∈ N, we have n n −n n kDf |Es(p)k≤ Cλ , kDf |Eu(p)k≤ Cλ . A periodic point p of period π is hyperbolic if it is a hyperbolic fixed point of f π. Smale generalized this notion to a compact invariant non-periodic set Λ, using the idea of “movement of the base point” (see [27]). Precisely, Λ is called hyperbolic s u if there are C > 0, 0 <λ< 1 and a continuous splitting TΛM = E ⊕ E , such that for any x ∈ Λ and n ∈ N, we have • Invariant property: Df(Es(x)) = Es(f(x)), Df(Eu(x)) = Eu(f(x)); • Exponential property: n n −n n kDf |Es(x)k≤ Cλ , kDf |Eu(x)k≤ Cλ . 2000 Mathematics Subject Classification. Primary: 37D20, 37D30; Secondary: 37D25. Key words and phrases. homoclinic class, hyperbolic time, shadowing lemma. 1143 1144 CHRISTIAN BONATTI, SHAOBO GAN AND DAWEI YANG Hyperbolic invariant sets have many nice properties, such as the stability proper- ties, the existence of stable manifolds, the shadowing lemma, etc.. By the shadowing lemma, the periodic orbits are dense in an isolated hyperbolic set with recurrence, and all the periodic orbits are hyperbolic. Here we investigate the following question: Problem. Can we obtain the hyperbolicity of an invariant compact set K by using the hyperbolicity of the periodic orbits in the set? In general, the hyperbolicity of the periodic orbits contained in K is not enough to get the hyperbolicity of K. For example, Kupka-Smale theorem asserts that every periodic orbit of Cr-generic diffeomorphisms is hyperbolic and that the invariant (stable and unstable) manifolds of the periodic orbits are pairwise transverse. How- ever there are open set O⊂ Diffr(M) of transitive diffeomorphisms f (in particular Ω(f)= M) such that f is not globally hyperbolic (see [28] for the first examples): generic diffeomorphisms in O are transitive and Kupka-Smale but not hyperbolic. Nevertheless there are some results where the global hyperbolicity can be ob- tained from the information on periodic orbits. • If the hyperbolicity of the periodic orbits is robust, that is, if C1 perturba- tions cannot create non-hyperbolic periodic orbits, then it implies the global hyperbolicity. More precisely, a diffeomorphism f is called C1 star if there is a C1 neigh- borhood U of f such that for any g ∈U, every periodic orbit of g is hyperbolic; it is proved by [16, 17, 18, 1, 14] that if f is C1 star, then f is Axiom A (i.e. the non-wandering set Ω(f) is a hyperbolic set and is the closure of the set of periodic points of f); • it is proved by [18] that for any C2 one-dimensional endomorphism f, for a compact invariant set Λ of f, if every periodic point in Λ is hyperbolic, then either Λ is hyperbolic, or f is topologically equivalent to an irrational rotation; • The one-dimensionnal result of Ma˜n´ehas been generalized to diffeomorphisms on compact surfaces in [25] by assuming the existence of a dominated splitting. Let Λ be an invariant compact set of a C2 diffeomorphism f of a compact surface. If Λ has a dominated splitting and every periodic point in Λ is a hyperbolic saddle, then Λ=Λ1 ∪ Ci, 1≤[i≤n where Λ1 is hyperbolic, and each Ci is a periodic normally hyperbolic circle ni with period ni such that f |Ci is conjugated to an irrational rotation. 1.2. Hyperbolicity of a homoclinic class versus existence of weak periodic points in the class. The aim of this work is to show that the hyperbolicity of a homoclinic class can be obtained from the periodic orbits contained in it (for C1 diffeomorphism and without perturbation). Let us first define the homoclinic classes which are the main object of our study. Given two hyperbolic periodic points p and q, we say they are homoclinically related if W s(Orb(p)) ⋔ W u(Orb(q)) 6= ∅ and W u(Orb(p)) ⋔ W s(Orb(q)) 6= ∅, and is denoted by q ∼ p. If p and q are homoclinically related, they must have the same index (i.e., the dimension of the stable manifold). This relation is an equivalence relation in the set of hyperbolic periodic points, and the homoclinic class of a hyperbolic periodic point p is defined by HYPERBOLICITY OF HOMOCLINIC CLASSES 1145 H(p)= {q : q ∈ P (f), q ∼ p}; A useful characterization of the homoclinic class is H(p)= W s(Orb(p)) ⋔ W u(Orb(p)). The homoclinic class of p is a transitive compact invariant set associated to a peri- odic point p. The homoclinic classes play an important role for the organization of the global dynamics using elementary pieces: • For Axiom A diffeomorphisms, the homoclinic classes are Smale’s basic pieces. • For homeomorphisms on compact metric spaces, Conley theory splits the chain recurrent set in chain recurrence classes. According to [4], for C1-generic dif- feomorphisms the homoclinic class H(p) is the chain recurrent class containing p. In general a non-trivial homoclinic class H(p) is not hyperbolic although it con- tains many non-trivial hyperbolic invariant sets: any finite set of periodic orbits homoclinically related with p is contained in a hyperbolic basic set. However, H(p) may contain periodic points which are not homoclinically related with p. Actually, the indices of the periodic orbits in H(p) may be any subset like {i,i +1, · · · , j}. Our main result will show that, some conditions on the periodic orbits homo- clinically related with p imply the hyperbolicity of H(p). Now we will introduce precisely the notions for stating these conditions. Definition 1.1. Let Λ be a compact invariant set of a diffeomorphism f. Let TΛM = E ⊕ F be a splitting of the tangent bundle of M over Λ in direct sum of two subbundles. 1. The splitting TΛM = E ⊕ F is a dominated splitting if the bundles E and F are invariant under Df and if there are two constants C > 0 and 0 <λ< 1 such that for any x ∈ Λ and n ∈ N, we have n −n n kDf |E(x)k·kDf |F (f n(x))k≤ Cλ . We sometimes use the notation E ⊕< F for indicating that E is dominated by F . 2. The bundle E of a dominated splitting is called contracting, if there are C > 0, n n 0 <λ< 1, such that for any x ∈ Λ and n ∈ N, we have kDf |E(x)k≤ Cλ . 3. If furthermore the dimension dim(E(x)) is independent of x, then this domi- nated splitting is call homogenous and dim(E) would be called its index. 4. Moreover, if E is contracting, a compact neighborhood U of Λ is called admis- n sible if the maximal invariant set M(f,U)= ∩n∈Zf (U) has one and exactly ˆ ˆ ˆ one homogeneous dominated splitting TM(f,U)M = E ⊕F with E contracting, extending the dominated splitting TΛM = E ⊕ F . One easily verifies that dominated splittings are always continuous and admits admissible neighborhoods (see [7, Appendix B] for elementary properties of domi- nated splittings). A splitting TΛM = E1 ⊕E2 ⊕···⊕Ek in k invariant subbundles is a dominated splitting if each splitting TΛM = (E1 ⊕···⊕ Ei) ⊕ (Ei+1 ⊕···⊕ Ek) in two bundles is dominated. A dominated splitting TΛM = E1 ⊕ E2 ⊕···⊕ Ek is a partially hyperbolic splitting if E1 is contracted or Ek is expanded. Definition 1.2. Let {pn}n∈N be a sequence of hyperbolic periodic points of a s u diffeomorphism f with period π(pn), and E (pn), E (pn) their stable and unstable spaces (respectively). 1146 CHRISTIAN BONATTI, SHAOBO GAN AND DAWEI YANG u We say that f is uniformly E -expanding at the period on {pn}n∈N if there are real numbers K > 1 and 0 <λ< 1 and an integer k > 0, such that for any n ∈ N one has π(pn) −k π(pn) u j k(Df )|E (f (pn))k≤ Kλ . (#) jY=1 −π(pn) π(q) u Remark 1. The equality (#) is different from kDf |E (pn)k≤ Kλ . One easily verifies: π(pn) −kπ(pn) −k u u j kDf |E (pn)k≤ k(Df )|E (f (pn))k.
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