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 uniformly 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 properties, 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 obtained from the information on periodic orbits. • If the hyperbolicity of the periodic orbits is robust, that is, if C1 perturbations 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 neighborhood 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 Mañéhas 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 periodic 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 diffeomorphisms 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 contains 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 homoclinically 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 dominated 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 dominated 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. jY=1 The inequality is an equality if dimEu = 1. We are now ready for stating our main result: Main Theorem. Let p be a hyperbolic periodic point of a diﬀeomorphism f on a compact manifold. Assume that its homoclinic class H(p) admits a (homogeneous) dominated splitting TH(p)M = E ⊕ F with E contracting and dim(E) = ind(p). If f is uniformly F -expanding at the period on the set of periodic points q homoclinically related to p, then F is uniformly expanding on H(p). This theorem provides a criterion for getting “weak periodic points” in a given homoclinic class (where weak periodic points means that they have a Lyapunov exponent arbitrarily close to 0), i.e., Corollary 1. Let p be a hyperbolic periodic point of a C1-diﬀeomorphism f such that:

• the homoclinic class H(p) admits a partially hyperbolic splitting TH(p)M = Es ⊕ Ec ⊕ Eu with Es uniformly contracted, Eu uniformly expanded and dim Ec =1. • the index of p is dim Es. • H(p) is not hyperbolic. Then, for every ε > 0 there is a periodic point q ∈ H(p) homoclinically related to p and such that 1 π(q) log(|Df | c |) ≤ ε. π(q) E (q) 1.3. Hyperbolicity versus approximation of a homoclinic class by periodic orbits with another index. Main Theorem and Corollary 1 ensure the existence of weak periodic points in a given non-hyperbolic homoclinic class H(p); this has important consequences on the dynamical properties one can produce in the neighborhood of H(p) by C1-small perturbation. Let us recall a result on linear algebra due to Mañé[17] which allows us to change the index of weak periodic point. Consider a set of hyperbolic periodic orbits u X = n∈N Orb(pn) of a diffeomorphism f and assume that f is not uniformly E expandingS at period on X; then, arbitrarily small perturbation of the differential Df|Eu along one of the periodic orbits allows us to increase by 1 the index of one of the periodic point pn. Furthermore, by a Lemma of Franks, this change of the 1 differential of f along the orbit of pn can be performed by a C -small perturbation of f supported in an arbitrarily small neighborhood of the orbit. HYPERBOLICITY OF HOMOCLINIC CLASSES 1147

The weak periodic points in H(p) provided by Main Theorem and Corollary 1 are homoclinically related with p. Then, H(p) is the Hausdorff limit of such weak periodic points. Together with Mañé’s result, this implies as a direct corollary: Corollary 2. Let p be a hyperbolic periodic point of a C1-diffeomorphism f such that:

• the homoclinic class H(p) admits a partially hyperbolic splitting TH(p)M = Es ⊕ Ecu with Es uniformly contracted; • the index of p is dim Es; • H(p) is not hyperbolic. 1 Then, there is a sequence of diffeomorphisms fn converging to f in the C - topology such that fn has a hyperbolic periodic point qn with index(qn)= index(p)+ 1 and the orbits Orb(qn) for fn converge to H(p) in the Hausdorff topology. 1.4. Bifurcation phenomena versus hyperbolicity. Hyperbolicity usually means stable. There are many bifurcation phenomena break the hyperbolicity. For instance, • A hyperbolic periodic point of index 0 is called a source, and a hyperbolic periodic point of index d is called a sink. If f has infinitely many sinks or sources, f is not hyperbolic. In other words, infinitely many sinks or sources are bifurcation phenomena. This can happen in many cases: one can see [21, 22, 23] for higher topology and [5] for C1 topology. • We say f has a homoclinic tangency if there is a periodic point p such that W s(Orb(p)) intersect W u(Orb(p)) non-transversely at some point x. For these one can see [21, 22, 23, 26, 2]. In [25, 30, 13], it is shown that , if the stable/unstable splitting on a periodic orbit is not enough dominated, then one can get a homoclinic tangency associated to this periodic orbit by performing an ε-C1 perturbation in an arbitrarily small neighborhood of the orbit. As a direct consequence of this result with Corollary 2 one gets: Corollary 3. Let p be a hyperbolic periodic point of a C1-diffeomorphism f such that:

• the homoclinic class H(p) admits a partially hyperbolic splitting TH(p)M = Es ⊕ Ecu with Es uniformly contracted; • the index of p is dim Es; • H(p) is not hyperbolic. Then, • either one can split Ecu in Ecu = Ec ⊕ F such that dim(Ec)=1 and the splitting Es ⊕ Ec ⊕ F is dominated over H(p); 1 • or there is a sequence of diffeomorphisms fn converging to f in the C -topology such that fn has a hyperbolic periodic point qn with index(qn)= index(p)+1 and fn has an homoclinic tangency associated to qn ; furthermore the orbits Orb(qn) for fn converge to H(p) in the Hausdorff topology. For two-dimensional generic diffeomorphism f, Mañé[17] proved either f is Ax- iom A, or f has infinitely many sinks, or f has infinitely many sources. In three- dimensional case, we have some kinds of extension of this result. Corollary 4. If dim M =3, for a homoclinic class H(p), then one of the following possibilities follows: 1148 CHRISTIAN BONATTI, SHAOBO GAN AND DAWEI YANG

• H(p) is hyperbolic; • H(p) is partially hyperbolic with one-dimensional central bundle; • for any neighborhood U of f, there is g ∈ U such that g has a homoclinic tangency; • for any neighborhood U of f, there is g ∈ U such that g has infinitely many sinks or sources. 1.5. Main Theorem and a previous result by Mañé. The C1-stability conjecture is deeply associated to the name of Liao and Mañé. For surface diffeomorphisms, Liao’s selecting lemma [16] and Mañé’s ergodic closing lemma [17] play an important role, and they solved the stability conjecture for surface diffeomorphisms independently. For higher dimensional diffeomorphisms, Mañé[19] proves the stability conjecture, in his proof he used an elegant lemma whose proof is based on selecting periodic quasi-hyperbolic pseudo-orbits. The proof of Main Theorem is based on modifying the proof of Mañé’s lemma, but cannot be induced by Mañé’s lemma directly. We will introduce Mañé’s lemma now. Let Λ be a compact invariant set of a diffeomorphism f; assume that Λ admits s s a partially hyperbolic splitting TΛM = E ⊕ F , where E is uniformly contracting. For λ ∈ (0, 1) we say that a point x ∈ Λ is λ-inf expanded, if n 1 −1 lim inf log k(Df )|F (f j (x))k≤ log λ. n→+∞ n Xj=1 Here is a slightly stronger version of Mañé’s lemma: Lemma 1.3 ([19]). Let Λ be a compact invariant set of a diffeomorphism f. Assume that Λ is non-wandering in the sense that Ω(f|Λ)=Λ, where Ω(f|Λ) is the non- wandering set of f|Λ. Assume furthermore that Λ admits a homogeneous partially s s hyperbolic splitting TΛM = E ⊕ F , where E is contracting and dominated by F . If there is λ ∈ (0, 1) such that the set λ-inf expanded points is dense in Λ, then • either F is expanding (hence Λ is a hyperbolic set) • or, for every admissible neighborhood U of Λ, for any two numbers γ1 and γ2 such that λ<γ1 <γ2 < 1, there is a periodic point x in the maximal invariant n set M(f, U)= n∈Z f (U) with arbitrarily large period N satisfying T N N −1 N γ1 ≤ k(Df )|Fˆ(f j (x))k≤ γ2 . jY=1 The original proof of Mañéis difficult to read, and some places in the proof are unclear for us. Recently, [33] and [20] have given new complete proofs of this lemma. Although we can confirm Mañé’s lemma is true, the proof remains difficult. In this paper, we point out the main idea of this remarkable lemma in the proof of Main Theorem. Let us explain why Mañé’s lemma does not imply directly Main Theorem. These two results are not perturbation lemmas and they have no genericity hypotheses on the diffeomorphism. Hence the chain recurrence class of p can be larger than the homoclinic class H(p); the homoclinic classes of periodic points with the same index may have a non-trivial intersection without being equal (in that case the cor- responding points are not homoclinically related). Trying to apply Mañé’s lemma on H(p) for proving Main Theorem one gets “weak periodic points” in the neighborhood of H(p). Even if these periodic points were all homoclinically related, one HYPERBOLICITY OF HOMOCLINIC CLASSES 1149 gets that the homoclinic class of the weak points intersects H(p), but we did not find a way for proving that there are indeed homoclinically related with p.

Remark 2. Mañé’s lemma is used for C1-star diffeomorphisms and flows, see [11, 14, 29] for instance. We notice that the star condition is a robust property. The condition (#) is not a robust property. In Main Theorem, we don’t perturb f, and we don’t assume any robust property.

2. Preliminaries: Hyperbolic times, uniform points and obstruction points. In this section, we will introduce some important tools and some elementary but useful lemmas for the proof of Main Theorem. Let f be a diffeomorphism on a compact manifold M. For every x ∈ M and every n ≥ 0 we denote by (x, f n(x)) the segment of orbit (x, f n(x)) := {x, f(x), · · · ,f n(x)}. We consider a compact invariant set Λ which admits a partially hyperbolic split- s s s ting TΛM = E ⊕ F , for which E is dominated by F and E is contracting. We fix an admissible compact neighborhood U of Λ and one denotes by M(f,U)= i i∈Z f (U) the maximal invariant set in U. By definition of admissible, the splitting ETs ⊕ F extends in a unique way in a partially hyperbolic splitting on M(f,U).

2.1. Uniform strings and shadowing lemma. One of our main tools will be the uniform strings, which are deﬁned below:

Deﬁnition 2.1. For every γ ∈ (0, 1) and every x ∈ M(f,U), one says that the orbit segment (x, f n(x)), n ≥ 1 is a γ-string if

n −1 n k(Df )|F (f j (x))k≤ γ . jY=1

One says that (x, f n(x)) is a uniform γ-string if (f k(x), f n(x)) is a γ-string for any k ∈ [0, n).

The importance of the uniform strings is given by the shadowing lemma for quasi-hyperbolic pseudo-orbits.

Deﬁnition 2.2. For δ > 0, a sequence {x1, x2,...,xN }, N > 1, of points in M n is δ-shadowed by a periodic orbit x if N is a period of x and d(f (x), xn) < δ for 1 ≤ n ≤ N.

We will see that pseudo orbits built with uniform strings, whose extremities are very close, can be shadowed.

Deﬁnition 2.3. For ǫ > 0 and γ ∈ (0, 1), a sequence of uniform γ-strings ni k {(xi, f (xi))}i=1 is called a periodic (ǫ, γ)-quasi-hyperbolic pseudo-orbit if ni d(f (xi), xi+1)<ǫ for all 1 ≤ i ≤ k with xk+1 = x1.

Lemma 2.4 (Generalized Shadowing Lemma, [19], [10]). For δ > 0 and γ ∈ (0, 1), there is ǫ = ǫ(δ, γ) such that for any (ǫ, γ)-quasi-hyperbolic pseudo-orbit ni k {(xi, f (xi))}i=1, it can be δ-shadowed by a periodic point x. 1150 CHRISTIAN BONATTI, SHAOBO GAN AND DAWEI YANG

2.2. Existence of uniform strings: Pliss lemma. Pliss lemma ensures the existence of uniform strings:

Lemma 2.5 (Pliss Lemma [24]). For all 0 <γ1 <γ2 < 0, there is c(γ1, γ2) ∈ (0, 1) n such that for n ≥ 1, if (x, f (x)) is a γ1-string, then there exist positive integers ni 0 < n1 < n2 < · · · < nk ≤ n, k ≥ nc(γ1, γ2) such that (x, f (x)) is a uniform γ2-string for all 1 ≤ i ≤ k. For a simple proof of Pliss lemma we refer to [3, Lemma 3.1] Deﬁnition 2.6 (hyperbolic time). For a point x ∈ Λ and γ ∈ (0, 1), n ∈ N is called γ-hyperbolic time if (x, f n(x)) is a uniform γ-string. The set of all γ-hyperbolic times is denoted by HT (x, γ) and n-th γ-hyperbolic time is denoted by HTn(x, γ).

Thus, for 0 <γ1 <γ2 < 1, if a point x ∈ Λ is γ1-inf expanded (for deﬁnition see the introduction), then it admits inﬁnitely many γ2-hyperbolic times.

2.3. Uniform points and large unstable manifolds. Deﬁnition 2.7. Recall U is an admissible neighborhood of Λ. A point x ∈ n M(f,U)= ∩n∈Zf (U) is called γ-uniform point, if for any n ∈ N

n−1 −1 n k(Df )|F (f −j (x))k≤ γ . jY=0 In other words, for every n > 0, the orbit segment (f −n(x), x) is a γ-uniform string; equivalently n is a γ-hyperbolic time for f −n(x). The importance of γ-uniform point is given by the fact that they have well deﬁned unstable manifolds of a uniform size; more precisely, for every 0 <γ< 1, there is ρ˜γ > 0 such that, for every γ-uniform point x ∈ M(f,U), the unstable manifold u W (x) contains a disk of radiusρ ˜γ (this uniform sizeρ ˜γ just depends on γ, f and the domination) (see for example [25, Corollary 3.3] and [31, Lemma 3.8]). As a consequence one gets:

Corollary 5. For every 0 <γ< 1, there is δγ > 0 such that, for every γ-uniform points x, y ∈ M(f,U) one has u s d(x, y) <δγ =⇒ W (x) ⋔ W (y) 6= ∅. In particular, if x and y are periodic points, then they are hyperbolic, have the same index and are homoclinically related. 2.4. Uniform points in periodic orbits. The existence of uniform point is sim- pler on periodic orbits. Let x be a periodic point with period π(x), if for any 1 ≤ n ≤ π(x)

n−1 −1 n k(Df )|F (f −j (x))k≤ γ , jY=0 then x is an γ-uniform point.

Lemma 2.8. If a periodic point x is γ1-inf expanded, then there is y ∈ Orb(x) such that y is a γ2-uniform point for any γ2 >γ1. HYPERBOLICITY OF HOMOCLINIC CLASSES 1151

Proof. Since x is γ1-inf expanded, Pliss lemma implies that #HT (x, γ2) = ∞. Hence there are γ2-hyperbolic times n such that n > π(x). Thus, k k −1 − +1 k(Df )|F (f n j (x))k≤ γ2 , ∀k =0, 1, · · · , π(x) − 1. jY=0 n n−π(x) n From the fact f (x)= f (x), we get that f (x) is a γ2-uniform point. Remark 3. Indeed the proof of Lemma 2.8 implies that, for every periodic point n x, if y = f (x) with n > π(x) and n is a γ2-hyperbolic time of x, then y is a γ2-uniform point.

Corollary 6. Let x be a periodic point of period π(x) and assume that x is γ1-inf expanded. Then for every γ2 > γ1 there is a γ2 hyperbolic time of x in [1, π(x)]. i Furthermore, if i is the largest γ2-hyperbolic time in [1, π(x)] then f (x) is a γ2- uniform point.

Proof. By Lemma 2.8 we know that there is a point y ∈ Orb(x) which is a γ2- uniform point. We can write y = f j(x) with j ∈ [1, π(x)] showing the ﬁrst claim of the corollary. Consider now the largest γ2-hyperbolic time i ∈ [1, π(x)]. In i particular j ≤ i. Let us denote z = f (x). We now prove that z is a γ2-uniform point. For every n> 0, n−1 −1 − n • if n ≤ i−j one has k=0 k(Df )|F (f k(z))k≤ γ2 , because i is a γ2 hyperbolic times; Q • if n ≥ i − j one has n−1 −1 − k=0 k(Df )|F (f k(z))k Q n−i+j−1 i−j−1 −1 − −1 − = k=0 k(Df )|F (f k(y))k · k=0 k(Df )|F (f k (z))k Q Q n−i+j i−j n ≤ γ2 γ2 = γ2 ,

because y is a γ2-unifom point. 2.5. Obstruction points. Between two consecutive hyperbolic times, the estima- tion should be “bad”, otherwise, there would be new hyperbolic time between the two consecutive hyperbolic times, this would be a contradiction. This simple fact (explained precisely in the following lemma) is very important in the proof of Main Theorem for building “bad” points. Lemma 2.9. Let ℓ and m be k-th and (k +1)-th elements in HT (x, γ)∪{0}, where k ∈ N ∪{0}. For any s ∈ [1, m − ℓ), we have s −1 s ℓ k(Df )|F (f j (y))k >γ where y = f (x). ♦ jY=1 Proof. We may assume m − ℓ > 1. Since there is no γ-hyperbolic time between ℓ and m, s = 1 is true for (♦); otherwise, ℓ +1 would be a γ-hyperbolic time. We prove (♦) by induction. We assume that (♦) is true for r, but fails to be true for r +1, for r +1

s −1 s k(Df )|F (f j (f ℓ(x)))k >γ . jY=1 From above two inequalities, we have (f ℓ(x), f s+1(x)) is a uniform γ-string. There- fore, s +1is a γ-hyperbolic time for x. We get a contradiction. Deﬁnition 2.10. • For x ∈ Λ and γ ∈ (0, 1], the orbit segment (x, f n(x)) is called a γ-obstruction if k −1 k k(Df )|F (f j (x))k≥ γ , ∀1 ≤ k ≤ n. jY=1 • the point x is called a γ-obstruction point if (x, f n(x)) is a γ-obstruction for any n ∈ N. The set of all γ-obstruction points is denoted by Λ(γ). From the deﬁnition, one can see that Λ(γ) is a compact set (not invariant). The following lemma ex- plains that we can jump from a point which has a large obstruction segment to an obstruction point.

Lemma 2.11. For any 0 <γ< 1 and for any ǫ > 0, there is N = N(γ; ǫ) such that if (x, f N (x)) is a γ-obstruction, then d(x, Λ(γ)) <ǫ. N Proof. Let denote by ΛN (γ) the set of points such that the segment (x, f (x)) is a γ-obstruction. Then Λ(γ) = N>0 ΛN (γ). This is a decreasing intersection of compact sets, so for N large enoughT ΛN (γ) is contained in the ǫ-neighborhood of Λ(γ). By a simple calculation, if we have an obstruction for a large segment in the orbit, we will get the ﬁrst hyperbolic time by using a long time. This is the following observation (this is essentially used in Lemma 3.2).

Lemma 2.12. For 0 <γ1 <γ2 < 1 and N ∈ N, there is an integer η such that, if m m > η and (x, f (x)) is a γ2-obstruction, then

HT1(x, γ1) − m>N

(recall that HT1(x, γ1) is the ﬁrst γ1-hyperbolic time of x). −1 Idea of the proof. The proof of this lemma just uses the fact that infx∈M kDf (x)k m −1 m m > 0. The product j=1 k(Df )|F (f j (x))k is larger than γ2 , since (x, f (x)) is a γ2 obstruction. AsQ γ2 > γ1, we need to wait some time before the product N −1 N j=1 k(Df )|F (f j (x))k can be smaller than γ1 , which is necessary for having an hyperbolicQ time.

3. Proof of Main Theorem: Selecting suitable quasi-hyperbolic orbits. The proof of Main Theorem consists in selecting periodic quasi-hyperbolic pseudo orbits. The generalized shadowing lemma ensures the existence of periodic orbits close to our pseudo-orbits. These periodic orbits will be “weak”, which will contradicts the inequalities in the assumptions of Main Theorem, but not too weak, so that we will estimate the size of their unstable manifolds ensuring that they are all homoclinically related with p. This kind of argument is typical from the selecting lemma of Liao [16] (see [32] for an available reference). We argue by contradiction assuming that • p is a hyperbolic periodic point; HYPERBOLICITY OF HOMOCLINIC CLASSES 1153

• the homoclinic class H(p) admits a partially hyperbolic splitting TH(p)M = Es ⊕ F with Es uniformly contracted, and dim Es is the index of p; • H(p) is not hyperbolic, that is F is not uniformly expanded; • f is uniformly F -expanding at the period on the set of periodic points homoclinically related with p; by definition this means that there are real numbers K > 1 and 0 <λ< 1 and an integer k> 0, such that for any n ∈ N and every periodic point q ∼ p one has π(q) −k π(p) k(Df )|F (f j (p))k≤ Kλ . jY=1 Up to replace f by f k, we can (and we do) assume that k=1 in the equation above, that is π(q) −1 π(p) k(Df )|F (f j (p))k≤ Kλ . (1) jY=1 We fix an admissible compact neighborhood U of H(p). Recall M(f,U) = i i∈Z f (U) is the maximal invariant set in U. By definition of admissible, the Tsplitting Es ⊕ F extends in a unique way in a partially hyperbolic splitting on M(f,U). 3.1. First step for proving of Main Theorem: building inductively obstruction points.

Lemma 3.1. There is a point b0 ∈ H(p), such that for any n ∈ N, we have n −1 i kDf |F (f (b0))k≥ 1. iY=1

In other words, b0 is a 1-obstruction point. Proof. The proof is by contradiction. Assume that for every x ∈ H(p) there is n(x) −1 n(x) > 0 with i=1 kDf |F (f i(x))k < 1. The same n holds in a neighborhood of x. Hence by compactnessQ of H(p), one can choose N ∈ N and α ∈ (0, 1), such that n(x) −1 N n(x) ≤ N for any x ∈ H(p), and i=1 kDf |F (f i(x))k < α . Now by splitting ev- i ery orbit segment (x, f n(x)) in segmentsQ of the form (f i(x),f n(f (x))(f i(x))) (up to some iterates whose number is less than N) one gets that F is uniformly expanding n −1 n N : i=1 kDf |F (f i(x))k < Cα for some constant C > 0 and for any n ∈ . Q In order to start building a quasi hyperbolic pseudo orbit with the weak expanding property, we want to build a next obstruction point from a pervious obstruction point by a pseudo orbit consisting in two uniform strings, starting with a long obstruction segment. This induction step will be given by the next Lemma:

Lemma 3.2. Under the assumptions of Main Theorem, for any numbers µ1, µ2 with λ < µ2 < µ1 ≤ 1, any µ1-obstruction point b1 ∈ H(p), any integer N ∈ N, any ε> 0 there are points x, y ∈ H(p) and b2 ∈ H(p) and integers N1,N2 with the following properties:

• b2 is a µ2-obstruction point; • the points x and y are homoclinically related with p; N1 N2 • the orbit segments (x, f (x)) and (y,f (y)) are uniform µ2-strings and y = f N1 (x); 1154 CHRISTIAN BONATTI, SHAOBO GAN AND DAWEI YANG

N1−1 • the orbit segment (x, f (x)) is a µ2-obstruction segment; • N1 ≥ N; N2 • d(b1, x) <ε and d(b2,f (y)) <ε; N2 • f (y) is a uniform µ2-point. The proof of Lemma 3.2 is the aim of this section. Proof. As H(p) is a homoclinic class and b1 ∈ H(p), there is a sequence of periodic points {pn}⊂ H(p), homoclinically related with p such that limn→∞ pn = b1. By our assumption on the points homoclinically related with p, the points pn are all λ-inf expanded. Hence they have inﬁnitely many µ2-hyperbolic times. We consider all these µ2-hyperbolic times for {pn}. Let us deﬁne

φ1(n) = max HT (pn, µ2) ∩ [1, π(pn)] and

φ2(n) = min HT (pn, µ2) ∩ (π(pn), ∞).

By Lemma 2.8 and Corollary 6 we know that φ1(n) and φ2(n) exist. Furthermore φ1(n) φ2(n) f (x) and f (x) are a γ2-uniform points. Recall that HT1(pn, µ2) denotes the ﬁrst hyperbolic time of pn. Clearly one has φ2(n) − φ1(n) ≥ φ2(n) − π(pn) ≥ HT1(pn,γ2).

Claim. HT1(pn, µ2) → ∞ as n → ∞. As a consequence φ2(n) − φ1(n) → ∞ as n → ∞. Proof. If this claim is not true, if necessary we can choose a subsequence, we can assume that there is C such that for any n ∈ N, we have HT1(pn, µ2) ≤ C. Take a subsequence if necessary again, we can assume that HT1(pn, µ2) is constant, say L. This means that for all pn, we have L −1 L i kDf |F (f (pn))k≤ µ2 . iY=1

Since lim pn = b1, we have L −1 L i kDf |F (f (b1))k≤ µ2 . iY=1

This contradicts that b1 is a “µ1-bad point”. Claim. For every n large enough, one has the following properties:

• d(b1,pn) <ε; φ1(n) • There is a µ2-obstruction point b2 such that d(b2,f (pn)) <ε • HT1(pn, µ2) >N • HT1(pn, µ2) < φ1(n) Proof. The unique non trivial items are item 2 and 4. Let us prove them. The proof of item 2 is that φ1(n) and φ2(n) are consecutive µ2-hyperbolic times of pn. φ1(n) φ2(n)−1 Hence according to Lemma 2.9 the orbit segment (f (pn),f (pn)) is a µ2-obstruction. As φ2(n)−φ1(n) tends to inﬁnity, item 2 follows from Lemma 2.11. Item 4 follows form Pliss lemma, which asserts that for π(pn) large enough pn has strictly more than 1 µ2-hyperbolic time in [1, π(pn)].

For n large enough (given by the previous claim), take x = pn, N1 = HT1(pn, µ2), HT1(pn,µ2) y = f (x) and N2 = φ1(n) − HT1(pn). By construction; N1 N2 • (x, f (x)) and (y,f (y)) are uniform µ2-strings. HYPERBOLICITY OF HOMOCLINIC CLASSES 1155

• d(x, b1) <ε and d(y,b2) <ε and b2 is a µ2-obstruction point. N2 φ1(n) • f (y)= f (pn) is a uniform µ2-point. • Since N1 = HT1(pn, µ2) is the ﬁrst µ2-hyperbolic time of x Lemma 2.9 implies N1−1 that the orbit segment (x, f (x)) is a µ2-obstruction.

Corollary 7. For every ε> 0, every λ<γ2 <γ1 < 1 and every N > 0, there is an integer k, and a (ε,γ1)-quasi hyperbolic closed pseudo orbit Γ(ε,γ1,γ2,N) composed by 2k γ1-uniform strings

n0 m1 n1 m2 (y0,f (y0)), (x1,f (x1)), (y1,f (y1)), (x2,f (x2)) . . . nk−1 mk . . . (yk−1,f (yk−1)), (xk,f (xk)), with the following properties:

• all the points yi, xi+1, 0 ≤ i < k are homoclinically related with p; mi−1 • for every i ∈ {1,...,k}, the orbit segment (xi,f (x)) is a γ2-obstruction segment; mi • for every i ∈{1,...,k − 1}, yi = f (xi); • for every i ∈{1,...,k} one has mi >N · ni−1; ni • for every i ∈{0,...,k − 1} the point f (yi) is a γ1-uniform point.

Proof. We chose an inﬁnite sequence {µi}i∈N such that γ2 < · · · < µi+1 < µi · · · < µ0 < γ1 < 1. By Lemma 3.1, we ﬁx a 1-obstruction point b. In particular b is a γ1 obstruction point. Hence Lemma 3.2 (applied to the constant µ0 < γ1 ensures the existence of periodic points x0 ∼ p, y0 ∼ p, numbers m0,n0 and a point b0 such that

• b0 is a µ0-obstruction point. m0 n0 • the orbit segments (x0,f (x0)) and (y0,f (y0)) are uniform µ0-strings and m0 y0 = f (x0). m0−1 • the orbit segment (x0,f (x0)) is a µ2-obstruction segment. n0 1 • d(b0,f (y0)) < 4 ε. n0 • f (y0) is a uniform µ0-point.

Then, applying Lemma 3.2 to the µi−1-obstruction point bi−1 and to the numbers µi < µi−1, one builds by induction a sequence of periodic points xi,yi homoclinically related to p, a point bi and numbers mi,ni satisfying

• bi is a µi-obstruction point. mi ni • the orbit segments (xi,f (xi)) and (yi,f (yi)) are uniform µi-strings and mi yi = f (xi). mi−1 • the orbit segment (xi,f (xi)) is a µi-obstruction segment. • mi ≥ Nni−1. 1 ni 1 • d(bi−1, xi) < 4 ε and d(bi,f (yi)) < 4 ε. ni • f (yi) is a uniform µi-point.

Let us summarize the properties of the inﬁnite sequence x0,y0, x1,...,xn,yn,... we have built. For every i ∈ N one has:

• all the points yi, xi are homoclinically related with p; mi ni • the strings (xi,f (xi)) and (yi,f (yi)) are γ1-uniform strings; mi−1 • the orbit segment (xi,f (x)) is a γ2-obstruction segment; mi • yi = f (xi); • one has mi+1 > Nni; ni • the point f (yi) is a γ1-uniform point; mi mi 1 • d(f (yi), xi+1) ≤ d(f (yi),bi+1)+ d(bi+1, xi+1) ≤ 2 ε. 1156 CHRISTIAN BONATTI, SHAOBO GAN AND DAWEI YANG

The sequence we build satisﬁes all the announced properties but one: it is not 1 closed. By compactness of H(p) there is i and k > 0 such that d(yi,yi+k) < 2 ε. Now the family of 2k orbit segments

ni mi+1 ni+1 mi+2 (yi,f (yi)), (xi+1,f (xi+1)), (yi+1,f (yi+1)), (xi+2,f (xi+2)) . . . ni+k−1 mi+k . . . (yi+k−1,f (yi+k−1)), (xi+k,f (xi+k)), satisfies all the announced properties. 3.2. End of the proof: shadowing with control of the invariant manifolds. We fix a sequence {εN } given by the shadowing lemma such that every (εN ,γ1)- 1 quasi hyperbolic closed pseudo orbits in H(p) can be N shadowed by true closed orbits. We fix λ<γ4 < γ3 < γ2 < γ1 < γ0 < 1. We denote ΓN = Γ(εN ,γ1,γ2,N) the (εN ,γ1)-quasi-hyperbolic pseudo orbit build by Corollary 7. We denote by ON 1 a true orbit of f which is N -shadowing the pseudo-orbit ΓN . We ends the proof of Main Theorem by proving the following lemma

Proposition 1. There is N0 such that, for every N ≥ N0, the orbit ON is homoclinically related with p and satisﬁes

−1 π(On) k(Df )|F (x)k≥ γ4 . x∈YON Proof of Main Theorem: Notice that Proposition 1 implies that, the periodic orbits homoclinically related with p does not satisfy the inequality (1), and contradict our hypotheses on the uniform F expansion at the period of the periodic points related with p. This contradiction ends the proof of Main Theorem. It remains to prove Proposition 1. This proof is the aim of all this section. We ﬁrst prove:

Lemma 3.3. There is N0 such that, for every N ≥ N0 the orbit ON is homoclinically related with p.

Proof. The quasi-hyperbolic pseudo orbits ΓN are union of γ1-uniform strings

n0,N m1,N n1,N m2,N (y0,N ,f (y0,N )), (x1,N ,f (x1,N )), (y1,N ,f (y1,N )), (x2,N ,f (x2,N )), − nkN 1 mkN ..., (ykN −1,N ,f (ykN −1,N )), (xk,N ,f (xkN ,N )), where the points xi,N and yi,N are periodic points homoclinically related with p. We split the orbit ON into orbit segments

n0,N m1,N n1,N m2,N (˜y0,N ,f (˜y0,N )), (˜x1,N ,f (˜x1,N )), (˜y1,N ,f (˜y1,N )), (˜x2,N ,f (˜x2,N )), − nkN 1 mkN ..., (˜ykN −1,N ,f (˜ykN −1,N )), (˜xk,N ,f (˜xkN ,N )), j j 1 j j 1 so that d(f (xi,N ),f (˜xi,N )) < N for j ∈{0,...,mi,N }, d(f (yi,N ),f (˜yi,N )) < N mi,N ni,N for j ∈{0,...,ni,N }, and f (˜xi,N )=˜yi,N , and ﬁnally f (˜yi,N )=˜xi+1,N . Notice that, for N large enough the orbit ON is contained in the admissible neighborhood U of H(p), hence in the maximal invariant set M(f,U). Just using the absolute continuity of kDf −1k one gets

Claim. For N large enough, and for all i ∈{1, ki,N }, the orbit segments mi,N ni−1,N (˜xi,N ,f (˜xi,N )) and (˜yi−1,N ,f (˜yi−1,N )) are γ0-uniform strings. As a direct consequence one gets

ni,N Claim. For N large enough, the points f (˜yi,N ) are γ0-uniform points. HYPERBOLICITY OF HOMOCLINIC CLASSES 1157

ni,N By construction of the pseudo orbits ΓN , for every N the points f (yi,N ) is a γ1-uniform point (hence a fortiori a γ0-uniform point). Furthermore, the γ0- uniform points have a uniform size of their unstable manifold (the size of the stable manifold is also uniform because Es is uniformly contracted on M(f,U)). On the ni,N ni,N other hand the distance d(f (yi,N ),f (˜yi,N )) tends to 0. As a consequence of Corollary 5 one gets

ni,N ni,N Claim. For N large enough the points f (yi,N ) and f (˜yi,N ) are homoclinically related. As a consequence the orbits ON are homoclinically related with p. Lemma 3.3 proved the ﬁrst claim of Proposition 1. The second claim is that the orbits ON (for large N) are γ4-weak. The reason is that the orbits ON ar shadowing the pseudo-orbits ΓN which are composed by “weak segments”. That is the meaning of our next lemma:

Lemma 3.4. For N large enough, and for any i ∈ {0,...,kN − 1} we have the obstruction inequality

ni,N mi+1,N −1 −1 ni,N +mi+1,N j j kDf |F (f (yi,N )k · kDf |F (f (xi+1,N )k≥ γ3 jY=1 jY=1 Proof. By Corollary 7 one has

mi+1,N −1 −1 mi+1,N −1 j kDf |F (f (xi+1,N )k≥ γ2 . jY=1 −1 Let denote K = infx∈H(p) kDf |F (x)k. Then we have

ni,N −1 ni,N −1 j mi+1,N kDf |F (f (yi,N )k≥ K and kDf |F (f (xi+1,N )k≥ K. jY=1

By hypothesis, mi+1,N >N ·ni,N (see Corollary 7). Hence we conclude the proof by noticing that, for N large

mi+1,N −1 ni,N +1 ni,N +mi+1,N γ2 K >γ3 .

We ends the proof of Proposition 1 by the following corollary:

Corollary 8. For N large enough one has, for every i ∈{0,...,kN − 1} one has:

ni,N +mi+1,N −1 ni,N +mi+1,N j kDf |F (f (˜yi,N )k≥ γ4 jY=1 As a consequence −1 π(On) kDf |F (x)k≥ γ4 . x∈YON Proof. We write

ni,N +mi+1,N ni,N mi+1,N (˜yi,N ,f (˜yi,N )) = (˜yi,N ,f (˜yi,N )) ∪ (˜xi+1,N ,f (˜xi+1,N )) Now the announced inequality follows directly from Lemma 3.4, form the uniform −1 continuity of the function kDf |F (x)k for x ∈ M(f,U), and from the fact that the 1 orbit ON is N shadowing the pseudo orbit ΓN . 1158 CHRISTIAN BONATTI, SHAOBO GAN AND DAWEI YANG

3.3. Proof of Corollary 1. In the assumption of Corollary 1 we now assume that c u s c u the bundle F splits in F = E ⊕ E so that TH(p)M = E ⊕ E ⊕ E is a partially hyperbolic splitting with dim Ec = 1. We assume dim Es is the index of p and H(p) is not hyperbolic. According to [12], we can assume that the metric is adapted to this partially hyperbolic structure. So Main Theorem implies that the bundle F is not uniformly expanded at the period on the set of periodic points homoclinically related with p. For any γ < 1 there is a periodic orbit O homoclinically related with p such that −1 π(O) kDf |F (x)k≥ γ . xY∈O Notice that −1 −1 π(O) π(O) kDf |F (x)k≥ γ = Df |Ec(q)

xY∈O for every q ∈ O. Hence choosing γ < 1 such that log(γ−1) < ε we conclude the proof of Corol- lary 1: q is a hyperbolic periodic point homoclinically related with p and

1 π(q) log Df |Ec(q) ≤ ε. π(q)

4. Consequences of the existence of weak periodic points in H(p). The aim of this section is the proof of the consequences of Main Theorem we stated in the introduction.

4.1. Approximation of H(p) by periodic orbits q with ind(qn) = ind(p)+1. The aim of this section is the proof of Corollary 2. It follows from Main Theorem by standard arguments which have been used by several authors. For this reason we just give an outline of the proof without giving the technical details. Let p be a hyperbolic periodic point of a C1-diﬀeomorphism f such that the homoclinic class H(p) is not hyperbolic but admits a partially hyperbolic splitting s cu s s TH(p)M = E ⊕ E with E uniformly contracted and ind(p) = dim E . According to Main Theorem, for every λ< 1 every k > 0 and K > 0 there is a periodic point q homoclinically related with p such that

π(q) −k π(q) k(Df )|F (f j (q))k > Kλ . (2) jY=1 Let denote by W(λ,k,K) the set of periodic points q ∈ H(p) homoclinically related with p for which the inequality (2) holds.

Lemma 4.1. For every λ < 1, k > 0, and K > 0, there is a sequence qn ∈ W(λ,k,K) such that the orbits orb(qn) converge to H(p) in the Hausdorﬀ topology.

Idea of the proof. Given two periodic orbits O1, O2 homoclinically related with p, there are sequences of periodic orbits which spend an arbitrarily large time in an arbitrarily small neighborhood of O1, then goes in finite (bounded) time in a arbitrarily small neighborhood of O2 and comes back in the neighborhood of O1 in finite time. The precise formalization of this idea have been state in [6] as the notion of transition. This allows us to show that given any q0 ∈ W(λ,k,K) any periodic point x ∈ H(p) homoclinically related with p and any ε> 0 there is a point q1 ∈ W(λ,k,K) such that the orbits of q0 and x are contained in the ε-neighborhood HYPERBOLICITY OF HOMOCLINIC CLASSES 1159 of the orbit of q1. One concludes by induction on the periodic orbits homoclinically related with p and considering a sequence εn tending to 0. Let us rewrite [17, Lemma II.5] in our context Lemma 4.2. For every ε> 0 there are a number λ ∈ (0, 1), an integer k > 0 and K > 0 such that for every point q ∈ W(λ,k,K) there is an ε-small perturbation {Ai}i∈{0,...,π(q)−1} of the linear cocycle induced by Df|F over the orbit of q such that the perturbed linear map at the period Aπ(q)−1 ◦···◦ A1 ◦ A0 is not an expansion: there is an eigenvalue whose modulus is less then 1. Remark 4. Here there is one difficulty: we announced that we can create periodic points of index ind(p)+1. So we need that the perturbed cocycle has exactly 1 real eigenvalue less than 1. Let us show how to get this property: • First observe that one can obtain that only one eigenvalue with modulus less than one. For getting that, consider the barycentric isotopy between Df|F and the perturbation A, and consider the first parameter for which the linear map at the period is not an expansion. For this perturbation, one or more eigenvalues have modulus equal to 1 and the others have modulus strictly larger than 1. A small extra perturbation allows us to get exactly one eigenvalue of modulus less than 1. However this eigenvalue may be a complex eigenvalue. • Now, for solving this difficulty one may use [4][Lemma 6.6] (see also [8][Proposition 3.7]) which allows us to turn the eigenvalues to be real for linear cocycle over periodic orbits with large period, without changing the other eigenvalues. So, given ε > 0, we fix λ ∈ (0, 1), k > 0 and K > 0 as in Mañé’s lemma. According to Lemma 4.1, we choose a sequence qn ∈ W(λ,k,K) such that the orbits of qn converge to H(p) in the Hausdorff topology. Then, one uses Franks lemma (see [9]): one can realize the ε-perturbation of 1 Df|F over the orbit of qn by an ε-C -perturbation fn of f with support in an arbitrarily small neighborhood of the orbit of qn. Now qn is a hyperbolic periodic point of index ind(qn) = ind(p)+1 for fn, and H(p) is the Hausdorff limit of the sequence of the fn-orbits of qn. This concludes the proof of Corollary 2. 4.2. Homoclinic tangency / 1-dimensional center bundle. The aim of this section is the proof of Corollary 3. Here, with the same assumption as in Corollary 2 we assume furthermore that the bundle F does not admits any splitting F = Ec ⊕G c s c with dim E = 1, such that the splitting TH(p)M = E ⊕ E ⊕ G is dominated. 1 Consider the diffeomorphisms fn converging to f in the C -topology, given by Corollary 2. Let qn be a periodic of fn with index ind(p) + 1, (also given by Corollary 2) such that the orbits of qn converge to H(p) in the Hausdorff topology. [25, 30, 13] assert that, • either, the stable/unstable splitting is dominated for the linear cocycle induce by Dfn over the union of the fn-orbits of qn, 1 • or, for every ε, there are n and an ε-C -perturbation gn of fn in an arbitrarily small neighborhood of qn, such that gn coincides with fn in a smaller neighborhood of qn (in particular the period and the index of qn for gn are the same as for fn) and there is an homoclinic tangency associated to qn for gn. This ends the proof of Corollary 3. 1160 CHRISTIAN BONATTI, SHAOBO GAN AND DAWEI YANG

4.3. Bifurcation phenomena and hyperbolicity: Proof of Corollary 4. Let M be a closed 3 manifold and f : M → M be a diﬀeomorphisms. Let p be a hyperbolic periodic point which is a saddle. So its index is 1 or 2. Up to considering f −1 one may assume that ind(p) = 1. According to [13], if H(p) does not admit a dominated splitting TH(p)M = E ⊕< F with dim E = 1, then arbitrarily small C1-perturbations allow us to create homoclinic tangencies associated to the point p. In that case, we have got the conclusion of Corollary 4.

So we may assume that H(p) admits a dominated splitting TH(p)M = E ⊕< F with dim E = 1. For any ergodic measure µ supported in H(p) let λE denotes the Lyapunov exponents of µ in the E direction. In other words

λ (µ)= log kDf| kdµ. E Z E Lemma 4.3. Let Λ be an invariant compact set of a manifold (in any dimension). Assume that Λ does not contain hyperbolic sources. Assume furthermore that Λ admits a dominated splitting TΛM = E ⊕< F with dim E = 1. Then either E is uniformly contracting or, for every neighborhood U of Λ, there is a sequence of 1 diffeomorphisms fn, converging to f in the C -topology, such that fn has infinitely many hyperbolic periodic sources whose orbits are contained in U Proof. According to [24] an ergodic measure whose Lyapunov exponent are all strictly positive is the Dirac measure associated to the orbit of a periodic hyperbolic source, and hence is not supported in Λ. One deduces that, for every ergodic measure µ supported on Λ one has λE (µ) ≤ 0. If the bundle E is not uniformly contracted and dim E = 1, a classical argument (see for instance [17, pages 521-522]) shows that there is an ergodic measure µ supported on Λ such that λE (µ) ≥ 0, hence λE (µ) = 0. In that case, Mañé’s ergodic closing lemma implies that there are diffeomorphismsg ñ converging to f in 1 the C topology and periodic orbits γn ofg ñ with the following properties:

• the supports of the γn converge in the Hausdorff topology to the support of µ (contained in Λ) • the Lyapunov exponents λE(γn) (where E denotes the continuation forg ñ of the f-invariant bundle E) is arbitrarily close to 0. An arbitrarily small extra perturbation (using Franks Lemma [9]) builds diffeomorphisms gn for which γn is a periodic orbit with λE (γn) = 0. An arbitrarily small perturbation of gn build diffeomorphisms hn having normally expanding periodic segments σn containing infinitely many sources (accumulating on non-hyperbolic periodic points); furthermore, the orbits of σn tends (with n) to the support of µ for the Hausdorff topology, hence are contained in U.

If E is not uniformly contracted, Lemma 4.3 provides the conclusion of Corol- lary 4. So we may now assume that E is uniformly contracting and we denote Es = E. If H(p) is not uniformly hyperbolic Corollary 3 implies that

• either f is limit of diﬀeomorphisms fn having homoclinic tangencies associated to periodic orbit in the neighborhood of H(p); In that case the conclusion of Corollary 4 is satisﬁed; • or the bundle F splits in the direct sum of two 1-dimensional bundles F = c s c E ⊕ G such that the splitting TH(p) = E ⊕ E ⊕ G is dominated. HYPERBOLICITY OF HOMOCLINIC CLASSES 1161

s c So we can assume that H(p) admits a dominated splitting TH(p) = E ⊕< E ⊕< G in three 1-dimensional bundles such that Es is uniformly contracting. Now Lemma 4.3 applied to f −1 implies that either f is the limit of diﬀeomorphisms admitting hyperbolic periodic sinks in a arbitrarily small neighborhood of H(p) or G is uniformly expanding: in both case, we proved the conclusion of Corollary 4.

Acknowledgments. We will thank L. Wen and S. Crovisier for useful discussion and suggestion. D. Yang thanks l’IMB for its warm hospitality. S. Gan is supported by NSFC (10531010) and MOST (2006CB805903). D. Yang is supported by CSC of Chinese Education Ministry and NSFC (10701043).

REFERENCES

[1] N. Aoki, The set of Axiom A diffeomorphisms with no cycles, Bol. Soc. Bras. Mat., 23 (1992), 21–65. [2] M. Asaoka, Hyperbolic sets exhibiting C1-persistent homoclinic tangency for higher dimen- sions, Proc. Amer. Math. Soc., 136 (2008), 677–686. [3] J. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., 140 (2000), 351–398. [4] C. Bonatti and S. Crovisier, Récurrence et généricité, (French), Invent. Math., 158 (2004), 33–104. [5] C. Bonatti and L. D´ıaz, On maximal transitive sets of generic diffeomorphisms, Inst. Hautes études Sci. Publ. Math., 96 (2002), 171–197. [6] C. Bonatti, L. Diaz and E. Pujals, A C1-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicty or infinitely many sinks or sources, Ann. of Math., 158 (2003), 355–418. [7] C. Bonatti, L. Diaz and M. Viana, “Dynamics Beyond Uniform Hyperbolicity. A Global Geo- metric and Probabilistic Perspective,” Encyclopaedia of Mathematical Sciences, 102, Math- ematical Physics, III. Springer-Verlag, Berlin, 2005. [8] C. Bonatti, N. Gourmelon and T. Vivier, Perturbations of the derivative along periodic orbits, Ergodic Theory Dynam. Systems, 26 (2006), 1307–1337. [9] J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc., 158 (1977), 301–308. [10] S. Gan, A generalized shadowing lemma, Discrete and Continuous Dynamical Systems, 8 (2002), 627–632. [11] S. Gan and L. Wen, Nonsingular star flows satisfy Axiom A and the nocycle condition, Invent. Math., 164 (2006), 279–315. [12] N. Gourmelon, Adapted metrics for dominated splittings, Ergodic Theory Dyn. Syst, 27 (2007), 1839–1849. [13] N. Gourmelon, Generation of homoclinic tangencies by C1 perturbations, preprint, No. 502, http://math.u-bourgogne.fr/IMB/prepub/pre07.html. [14] S. Hayashi, Diffeomorphisms in F 1(M) satisfy Axiom A, Ergodic Theory Dyn. Syst, 12 (1992), 233–253. [15] M. Hirsch, C. Pugh and M. Shub, “Invariant Manifolds,” Lecture Notes in Math., 583, Springer-Verlag, Berlin, 1977. [16] S. Liao, On the stability conjecture, Chinese Annals of Math., 1 (1980), 9–30. [17] R. Mañé, An ergodic closing lemma, Ann. of Math., 116 (1982), 503–540. [18] R. Mañé, Hyperbolicity, sinks and measure in one-dimensional dynamics, Comm. Math. Phys., 100 (1985), 495–524. [19] R. Mañé, A proof of the C1 stability conjecture, Inst. Hautes études Sci. Publ. Math., 66 (1988), 161–210. [20] Kazumine Moriyasu and Kazuhiro Sakai, A note on Mañé’s proof of the stability conjecture, Far East J. Dyn. Syst., 4 (2002), 97–106. [21] S. Newhouse, Nondensity of axiom A(a) on S2, Global Analysis (Proc. Sympos. Pure Math.), XIV, Berkeley, Calif. (1968), 191–202. Amer. Math. Soc., Providence, R.I., 1970. [22] S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9–18. [23] S. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Inst. Hautes études Sci. Publ. Math., 50 (1979), 101–151. [24] V. Pliss, On a conjecture due to Smale, Diff. Uravnenija., 8 (1972), 268–282. 1162 CHRISTIAN BONATTI, SHAOBO GAN AND DAWEI YANG

[25] E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Ann. of Math., 151 (2000), 961–1023. [26] J. Palis and M. Viana, High dimension diffeomorphisms displaying infinitely many periodic attractors, Ann. of Math., 140 (1994), 207–250. [27] S. Smale, Differentiable dynamical systems, Bull. Am. Math. Sci., 73 (1967), 747–817. [28] M. Shub, “Topological Transitive Diffeomorphisms in T 4,” Lecture Notes in Math., 206, Springer-Verlag, Berlin, 1971. [29] L. Wen, On the C1 stability conjecture for flows, J. Differ. Equations, 129 (1996), 334–357. [30] L. Wen, Homoclinic tangencies and dominated splittings, Nonlinearity, 15 (2002), 1445–1469. [31] L. Wen, Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles, Bull. Braz. Math. Soc. (N.S.), 35 (2004), 419–452. [32] L. Wen, The selecting lemma of Liao, Discrete Contin. Dyn. Syst., 20 (2008), 159–175. [33] Y. Zhang and S. Gan, On Mañé’s proof of the C1 stability conjecture, Acta. Math. Sin. (Engl. Ser.), 21 (2005), 533–540. Received October 2008; revised May 2009. E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]