Some results on perturbations to Lyapunov exponents

In this paper, we study two properties of the Lyapunov exponents under small perturbations: one is when we can remove zero Lyapunov exponents and the other is when we can distinguish all the Lyapunov exponents. The first result shows that we can perturb all the zero integrated Lyapunov exponents $\int_M \lambda_j(x)d\omega(x)$ into nonzero ones, for any partially hyperbolic diffeomorphism. The second part contains an example which shows the local genericity of diffeomorphisms with non-simple spectrum and three results: one discusses the relation between simple-spectrum property and the existence of complex eigenvalues; the other two describe the difference on the spectrum between the diffeomorphisms far from homoclinic tangencies and those in the interior of the complement. Moreover, among the conservative diffeomorphisms far from tangencies, we prove that ergodic ones form a residual subset.


Introduction
It was shown in the 1960s that uniformly hyperbolic systems are not dense in the space of dynamical systems. This brought about the naissance of the notions of partial and nonuniform hyperbolicity. Using the concept of Lyapunov exponents, Pesin theory of nonuniformly hyperbolic systems(characterized by all the Lyapunov exponents are non-null for some invariant measure) gives us a rich information about geometric properties of the system. In particular, the points with all Lyapunov exponents non-zero have well-defined unstable and stable invariant manifolds. These tools are the base of most of the results on dynamical systems nowadays. Thus it is of utmost importance to detect when the zero exponents can be removed by perturbations.
One central result in this direction for discrete system is the Shub-Wilkinson example [32]. They build a conservative perturbation to a skew product of an Anosov diffeomorphism of the torus T 2 by rotations and creates positive exponents in the center direction for Lebesgue almost every point. Baraviera-Bonatti present a local version of Shub-Wilkinson's argument, allowing one to perturb the sum of all the center integrated Lyapunov exponents of any conservative partially hyperbolic systems.
Highly inspired by their results we prove in the present paper that one can perturb every integrated Lyapunov exponent M λ j (x)dω(x) of any conservative partially hyperbolic systems to nonzero ones, not only the sum of them. This is a generalization to Baraviera-Bonatti [8].
Let M be a d−dimensional compact Riemannian manifold without boundary (d ≥ 2) and ω be a smooth volume form. Denote by Diff 1 ω (M ) the set of volume preserving C 1 −diffeomorphisms on M . Take f ∈ Diff 1 ω (M ). By Oseledec Theorem, there exists a Df -invariant decomposition converges uniformly on {ν ∈ E i (x) : ν = 1}, where 1 ≤ s(f, x) ≤ d. This number λ i (f, x) is called the Lyapunov exponent associated with E i . Lyapunov exponents describe the asymptotic evolution of tangent map: positive or negative exponents correspond to exponential growth or decay of the norm, respectively, whereas vanishing exponents mean lack of exponential behavior. None of these values depends on the choice of a Riemannian metric. Throughout this paper, let λ 1 (f, x) ≥ λ 2 (f, x) ≥ · · · ≥ λ d (f, x) be the Lyapunov exponents in nonincreasing order and each repeated with multiplicity dimE i (x). We define the j − th−integrated exponent by M λ j (f, x)dω(x), for any 1 ≤ j ≤ d.
A Df −invariant splitting T M = E 1 ⊕ · · · ⊕ E k is called a dominated splitting if each E i is a continuous Df −invariant subbundle of T M and if there is some integer n > 0 such that, for any x ∈ M , any i < j and any non-zero vectors u ∈ E i (x) and ν ∈ E j (x), one has Df n (u) u < 1 2 Df n (ν) ν .
Let PH 1 ω (M ) denote the subset of Diff 1 ω (M ) consisting of all the partially hyperbolic diffeomorphisms. In this paper, partially hyperbolic means the following. There is a nontrivial splitting of the tangent bundle, T M = E u ⊕ E c ⊕ E s , that is invariant under the derivative map Df . Further, there is a Riemannian metric for which we can choose continuous positive functions ν,ν, γ andγ with ν,ν < 1 and ν < γ <γ −1 <ν −1 such that, for any unit vector υ ∈ T p M , (1.4) Theorem 1.1. Let f ∈ PH 1 ω (M ).Then for any neighborhood U ⊆ PH 1 ω (M ) of f , there is a diffeomophism g ∈ U, such that every integrated Lyapunov exponent is different from zero, i.e., Much work on perturbation of Lyapunov exponents concerns two basic topics. One is when we can remove zero Lyapunov exponents as we discussed above. The other is when we can distinguish all the Lyapunov exponents. That is, the spectrum is simple. About the latter topic, all work are concentrated for cocycles. In the case of independent and identically distributed random matrices, Arnold and Cong [4] showed that the cocycles with simple Lyapunov spectrum form a residual set. In the space of all linear cocycles equipped with the uniform topology, Knill [25] (for the 2dimensional case) and Arnold and Cong [5] (for the general d−dimensional case) proved that they form a dense set. For SL(d, R)−cocycles, Bonatti and Viana [17] also showed the density property under some conditions. It is natural to ask whether the diffeomorphisms with simple spectrum form a dense set in Diff 1 ω (M ). In Section 3, we construct an example related to this problem and show the local genericity of diffeomorphisms with non-simple spectrum. Using the techniques developed in [18] and [26], we prove a dichotomy between simple-spectrum property and the existence of complex eigenvalues.
Hence, the multiplicity of every center Lyapunov exponent of f is 1; • or for any ε > 0, there is an ε−perturbation g ∈ PH 1 ω (M ) of f and a periodic point q of g such that Dg p(q) q has a complex center eigenvalue , where p(q) is the period of q.
In the previous theorem, the first case is that each invariant subbundle in the finest dominated splitting is 1-dimensional. For the center submanifold, this property is much stronger than that of simple spectrum. Results in [34][28] [27] indicates that, roughly speaking, dominated splitting and homoclinic tangency are mutually exclusive concepts. We prove two results which suggest that diffeomorphisms far from homoclinic tangencies and in the interior of the complement load bearing entirely different properties. Moreover, applying the novel approach developed by F.R.Hertz, M.A.R.Hertz, A.Tahzibi, and R.Ures [23,24] for Pugh-Shub's stable ergodicity conjecture, we point out the genericity of the ergodic diffeomorphisms in the conservative ones which are far from tangencies.
We say a diffeomorphism f has C 1 homoclinic tangency if f ∈ Diff 1 (M ) has hyperbolic periodic point p at which the stable and unstable invariant manifolds W s (p) and W u (p) intersect nontransversely. Denote by HT the set containing all the diffeomorphisms with homoclinic tangencies. We call a diffeomorphism f is far away from tangency if f ∈ Diff 1 (M ) \ HT .
In the context of the following two results, we suppose d = dimM ≥ 3.
There is a C 1 residual subset R of the volume preserving diffeomorphsims far from tangencies, such that any f ∈ R admits a finest dominated splitting E s ⊕ E c1 ⊕ · · · ⊕ E cc ⊕ E u , where dim(E ci ) = 1 and c = dimE c . Moreover, f is ergodic.
There is a C 1 residual subset R of the volume preserving diffeomorphsims far from tangencies, such that each center Lyapunov exponent of f ∈ R has multiplicity 1.
Let CPH 1 ω (M ) denote the set of all center bunching, conservative, partially hyperbolic diffeomorphisms. The statement of the third result about spectrum is the following: There is a C 1 residual subset of diffeomorphism R in CPH 1 ω (M ) ∩ int(HT ), such that for any f ∈ R, it is ergodic, and it has two center exponents equal.
2 Proof of Theorem 1.1 Given a vector space V and a positive integer p, let ∧ p (V ) be the p − th exterior power of V . This is a vector space of dimension C p d , whose elements are called p − vectors. It is generated by the p−vectors of the form ν 1 ∧ · · · ∧ ν p with ν j ∈ V , called the decomposable p−vectors. A linear map If V has an inner product, then we endow ∧ p (V ) with the inner product such that ν 1 ∧ · · · ∧ ν p equals the p−dimensional volume of the parallelepiped spanned by ν 1 , · · · , ν p .
More generally, there is a vector bundle ∧ p (E), with fibers ∧ p (E x ), associated to E, and there is a vector bundle automorphism ∧ p (F ), associated to F . If the vector bundle E is endowed with a continuous inner product, then ∧ p (E) also is. The Oseledec data of ∧ p (F ) can be obtained from that of F , as shown by the proposition below.
Let {e 1 (x), · · · , e d (x)} be a basis of E x such that Then the Oseledec space E j,∧ p x of ∧ p (F ) corresponding to the Lyapunov exponentλ j (x) is the sub-space of ∧ p (E x ) generated by the p − th vectors e i1 ∧ · · · ∧ e ip , with 1 ≤ i 1 < · · · < i p ≤ d and λ i1 + · · · + λ ip =λ j (x).
A dominated splitting E 1 ⊕ · · · ⊕ E n is called the finest dominated splitting if there is no dominated splitting defined over all M in each invariant bundle E i , 1 ≤ i ≤ n. The continuation of the finest dominated splitting may not be the finest dominated splitting of the perturbation diffeomorphism. However, we can perturb any partially hyperbolic diffeomorphism to obtain a diffeomorphism with robust finest dominated splitting.
Lemma 2.2. The diffeomorphisms with robust finest dominated splitting are C 1 dense among the C r , r ≥ 1, partially hyperbolic diffeomorphisms of M .
proof For any neighborhood U of f in PH 1 ω (M ), we denote U f ⊂ U be the neighborhood of f in PH 1 ω (M ) such that the continuation of the dominated splitting of f is a dominated splitting of the perturbation diffeomorphism. Assume that there is a small perturbation h 1 ∈ U f such that the continuation of the finest dominated splitting of f is not the finest one of h 1 . Then the finest dominated splitting of h 1 must be a refinement of the continuation. Consider the subset U h1 ∩ U f . If there exists another perturbation h 2 in U h1 ∩ U f whose finest dominated splitting is a refinement of the continuation of the finest dominated splitting of h 1 , we shift our attention to a smaller subset U h2 ∩ U h1 ∩ U f . The process will stop at some perturbation h n since the number of invariant bundles in the finest dominated splitting of h i strictly increases ( as i increases ) and this number can not exceed d, the dimension of M . Let g = h n . Then g must has robust finest dominated splitting.
The main techniques in the proof of Theorem 1.1 are three results as we cite in the following: A partially hyperbolic diffeomorphism f is accessible if, for every pair of points p, q ∈ M , there is a C 1 path from p to q whose tangent vector always lies in E u ∪ E s and vanishes at most finitely many times. We say f is stably accessible if every g sufficiently C 1 −close to f is accessible. [21]). For any r ≥ 1, stable accessibility is C 1 dense among the C r , partially hyperbolic diffeomorphisms of M , volume preserving or not.

Lemma 2.3 (Main Theorem in
Lemma 2.4 (Thereom 2 in [8]). Let M be a compact manifold and ω be a smooth volume form.
Then there are ω−preserving diffeomorphisms g, arbitrarily C 1 −close to f , for which the inte- and Γ We recall from Proposition 2.1 that for any diffeomorphism f ∈ Diff 1 ω (M ) and p ∈ {1, ..., d − 1}, the integrated Lyapunov exponent of ∧ p (Df ) coincides with the sum of the largest p integrated Lyapunov exponents, that is, Lemma 2.5 (Proposition 4.17 in [11]). Let f ∈ Diff 1 ω (M ) and p ∈ {1, ..., d − 1}, and given any Proof of Theorem 1.1 By Lemma2.3, we can perturb f to a diffeomorphism f 1 ∈ PH 1 ω (M ) with stable accessibility. Take a neighborhood U 1 ⊂ PH 1 ω (M ) of f 1 such that each diffeomorphism in U 1 is accessible. Applying Lemma2.2, we get another small perturbation f 2 ∈ U 1 which is accessible and has the robust finest dominated splitting: (Since any integrated Lyapunov exponent related to E s and E u is robustly away from zero, we need not care about the decomposition of E s and E u ). Therefore, there is a neighborhood U 2 ⊂ U 1 of f 2 such that any g ∈ U 2 , g is accessible and has the same finest dominated splitting as f 2 . By perturbing f 2 if necessary, Lemma2.4 ensures that there exists a neighborhood U 3 ⊂ U 2 with property: That is, the sum of the integrated Lyapunov exponents where we denote s = dimE s and u = dimE u and c i = dimE ci and c 0 = 0 for simplicity. Notice that Hence the continuity points of the map form a residual subset. We choose a continuity point h of the above map in U 3 . Now we verify that h meets the requirements of our theorem.
Since h ∈ U 3 ⊂ U 1 , h is accessible. This implies that for ω−almost every x ∈ M , the orbit of x is dense in M . In fact, Burns-Dolgopyat-Pesin [18] pointed out in the proof of Theorem 2 that the essential accessibility property indicates that almost every point has a dense orbit. Note that the essential accessibility property is a weaker property than accessibility. Precisely, recall that accessibility is an equivalence relation. If a diffeomorphism is accessible then the partition into accessibility classes is trivial. A diffeomorphism is said to be essentially accessible if the partition into accessibility classes is ergodic (i.e. a measurable union of equivalence classes must have zero or full measure). From the definition, one can deduce that an accessible diffeomorphism is also essentially accessible. Hence h is essentially accessible, since h is accessible. And then almost every point has a dense orbit.
Then for any s In fact, if one has that ω(D p (h, m)) > 0 for some integer m > 0, by the accessibility, for ω−almost every point in D p (h, m), we can spread the dominated splitting along its orbit to the closure, that is, the whole manifold M . Since h has the same finest dominated splitting as f 2 , we have that Note that h is a continuity point of map (2.3). Combining with Lemma2.5, one can obtain that J p (h) = 0, i.e., Remark When we have done this paper, we find [23] which is considering the same problem in Theorem 1.1 of the case dimE c = 2.
3 The example and Proofs of Theorems 1.2-1.5 In this section we will first present an example of a linear Anosov diffeomorphism A on T 3 which has a couple of complex eigenvalues. Then we show that for any small neighborhood of A in Diff 1 ω (T 3 ), there exists a residual subset consisting of diffeomorphisms with non-simple spectrum. Then we will give the proof of Theorems 1.2, 1.3 and 1.5.

The example
Then A is stable ergodic and Since f (1) = 1 > 0 and f (2) = −3 < 0, by the continuity of f , there is c ∈ (1, 2), such that On the other hand, one consider the converse matrix We have that g(0) = 1 > 0 and g(1) = −1 < 0. This implies that ∃d ∈ (0, 1), such that g(d) = 0. Note that g ′ (λ) = −3λ 2 − 1 < 0. Therefore, the point d is the unique real root of g(λ) and hence the other two eigenvalues are not real. Moreover, observing that the coefficients of the first and last items are both 1, we can deduce that d must be an irrational number. Note that when a number λ is an eigenvalue of A, its reciprocal 1 λ is an eigenvalue of A −1 and conversely, it also holds. Thus we have the following conclusion about A.
Conclusion I. ∃p ∈ T 3 (hyperbolic fixed point) has complex eigenvalues λ 1 , λ 1 and a real (irrational) eigenvalue λ 2 satisfying This conclusion implies that A has non-simple spectrum. Moreover, applying the following lemma, we will show that for any small neighborhood of A in Diff 1 ω (T 3 ), there is a residual subset composed of diffeomorphisms with non-simple spectrum. Lemma 3.1 (Theorem 1 in [11]). There exists a residual set R ⊂ Diff 1 ω (M ) such that, for each f ∈ R and ω−almost every x ∈ M , the Oseledec splitting of f is either trivial or dominated at x.
We choose a neighborhood U ⊂ Diff 1 ω (T 3 ) of A, such that ∀g ∈ U, g is ergodic and has a fixed point (i.e., the continuation of p ) with complex eigenvalues. Using Lemma3.1, there is a residual set R ⊂ U, such that ∀ g ∈ R, the Oseledec decomposition of g is dominated or trivial. Therefore, ∀ g ∈ R, if g has simple spectrum, its dominated splitting must be the form for every generic point x ∈ O(ω), the Oseledec basin of ω, since g is ergodic. Note that ω is a volume form and the Oseledec basin has ω−full measure, the Oseledec basin is dense in M . Thus the dominated splitting E 1 (x) E 2 (x) E 3 (x) of g can be extended to the whole manifold M by the continuity of dominated splitting. This contradicts with the property that g has complex eigenvalues. Thus we have the second conclusion about the local genericity of diffeomorphisms with non-simple spectrum.
Conclusion II. For any small neighborhood V ⊂ Diff 1 ω (T 3 ) of A, there exists a residual subset R ⊂ V consisting of diffeomorphisms with non-simple spectrum.

The proof of Theorem 1.2
Bonatti and Crovisier [12] proved that, generically, a volume-preserving diffeomorphism is transitive in a compact, connected manifold with a volume-preserving diffeomorphism. To prove Theorem 1.2, we need introduce some preliminary notions and lemmas first. • Σ is a set and f : Σ → Σ is a one-to-one map; • π : E → Σ is a linear bundle of dimension d over Σ, whose fibers are endowed with an Euclidean metric · . The fiber over the point x ∈ Σ will be denoted by E x ; We shall say that a linear cocycle A is bounded if there exists a constant K > 0 such that, for any x ∈ Σ, we have A x < K and A −1 x < K. We call K a bound of A .
Remark Here the base space Σ can be taken arbitrarily, even not a topology space.
Let (Σ, f, E, A) be a linear system, an invariant subbundle is a collection of linear subspaces F (x) ⊂ E x whose dimensions do not depend on x and such that A(F (x)) = F (f (x)). An A−invariant splitting F ⊕ G is given by two invariant subbundles such that E Definition 3.4. Let (Σ, f, E, A) be a linear system and E = F ⊕ G an A−invariant splitting. We say that F ⊕ G is a dominated splitting if there exists n ∈ N such that for every x ∈ Σ. We write F ≺ G.
If we want to emphasize the role of n then we say that F ⊕ G is an n−dominated splitting and write F ≺ n G. Finally, the dimension of the dominated splitting is the dimension of the subbundle F . The following is Theorem 3.6 in [26] with more details. Lemma 3.6 (Theorem 3.6 in [26]). Given K > 0 and ε > 0 there is ℓ ∈ N such that for any linear periodic system A = (Σ, f, E, A) bounded by K, one has that it admits a finest dominated splitting and only if we can not get a complex eigenvalue of the following ranks (τ 1 , τ 1 + 1), (τ 2 , τ 2 + 1), · · · , (τ k−1 , τ k−1 + 1) Proof of Theorem 1.2: Let G ω be the residual subset of Diff 1 ω (M ) determined by Lemma 3.2. Define R = G ω ∩ PH 1 ω (M ). Then R is a residual subset PH 1 ω (M ) of diffeomorphisms f such that M is the unique homoclinic class.
For any diffeomorphism f ∈ R, we take K = max x∈M { Df | E c , Df −1 | E c }. Then we obtain a positive integer ℓ by Lemma 3.6. If E c admits an ℓ−finest dominated splitting as the form (1.5), the proof is done. Otherwise, there is an integer s < i < s + c, such that E c (x) has no dominated decomposition E c (x) = E(x) ⊕ ≺ F (x) with dimE = i − s for x in some positive-measure subset of M . Then there are periodic points such that the cocycles defined over these points do not have dominated decomposition E c = E ⊕ ≺n F with dimE = i − s and uniform time n, for any integer n ∈ N.
Then A = (Σ, f, E, A) is a linear cocycle bounded by K over an infinite periodic system having transitions but without dominated splitting E c = E ⊕ ≺ F where dimE = i − s. Now using Lemma 3.6, for any ε > 0, we obtain a ε−pertubation B of A and a periodic point in Σ, such that M q,B = B(f p(q)−1 q) • · · · • B(q) has a complex eigenvalue of rank (i, i + 1). Applying Franks' Lemma, we get a C 1 −perturbation g ∈ PH 1 ω (M ) of f , which coincides with f out of an arbitrarily small neighborhood of Orb(q), equals to f in Orb(q), and whose derivation satisfying Hence Dg p(q) q has a complex eigenvalue.

The proof of Theorem 1.3
In this subsection, we focus on conservative diffeomorphisms far from homoclinic tangencies and discuss their two generic properties: one is the property of admitting a dominated splitting as (1.5) (Lemma 3.7) and the other is ergodicity (Lemma 3.10).
, it is C 1 far away from tangencies if and only if there exists a dominated splitting E s ⊕ E c ⊕ E u with two non-trivial extreme subbundles and the finest dominated splitting of the center bundle is of the form Abdenur-Bonatti-Crovisier-Diaz-Wen claim that, for C 1 -generic diffeomorphisms, the set of indices of the (hyperbolic) periodic points in a chain recurrence class (in fact, such classes are homoclinic ones) form an interval in N. Applying the connecting lemma and Franks' lemma for conservative diffeomorphisms, we obtain the conservative version of this result (Theorem 1.1 in [1]) as follows: Recall that a point x ∈ M is called (C 1 ) i−preperiodic of f , 0 ≤ i ≤ d, if for any neighborhood U of f in Diff 1 (M ) and any neighborhood U of x in M , there exist g ∈ U and y ∈ U such that y is a hyperbolic periodic point of g with index i. Now we begin to prove one of the main lemma in this subsection: where G ω and I are determined by Lemma 3.2 and 3.8, respectively. For f ∈ R, let i 0 (resp i 1 ) be the minimal (resp. maximal) preperiodic index of f . Then i 0 ≥ 0 and i 1 ≤ d. By the definition of preperiodic points, we can assume that f itself contains index i 0 , i 1 periodic points. By Lemma 3.8, f contains i−index periodic points for all i ∈ [i 0 , i 1 ] ∩ N. Since M is the unique homoclinic class by Lemma 3.2 and f is far from tangencies, now we obtain a dominated splitting with center bundles all 1 dimensional by the equivalent conditions for the existence of dominated splitting(for more details, see [28,34]). Since f is conservative, one must have that i 0 ≥ 1 and i 1 ≤ d − 1. Otherwise, if i 0 = 0, the minimal of periodic index can be 0 or 1(There may exist a weak stable subbundle with dimension 1). In the first case, all the Lyapunov exponents of f are non-negative. Hence the sum of the Lyapunov exponents are nonzero. This contradicts with the conservative condition. In the second case, the negative exponent have multiplicity 1 and is close to 0, but the positive exponents are uniformly far from 0 by the domination. This again deduces the same contradiction as the first case. The proof of i 1 ≤ d − 1 is analogous.
The rest thing is to prove the two extreme bundles are uniformly hyperbolic. This can be obtained by showing that it is forbidden to decrease the index of a periodic point with index i 0 or increase that of a periodic point with index i 1 by perturbation. The proof essentially from M añé's Ergodic Closing Lemma [29], in conservative version (see also [36] or Theorem B and Section 4 in [15] ). This concludes the proof of necessity. (G1) f is partially hyperbolic, it admits a partially hyperbolic splitting admits a finest dominated (G2) f has periodic points with index dim(E s ), dim(E s ) + 1, · · · , dim(E s ) + c(f ).
Recently, F.R.Hertz, M.A.R.Hertz, A.Tahzibi, and R.Ures [23,24] developed a new criteria of ergodicity and nonuniform hyperbolicity which provided fresh ideas to the Pugh-Shub stable ergodicity conjecture. Applying Proposition 3.9 and their criteria, we illustrate the density of ergodicity as the follow which implies the ergodic diffeomorphisms form a residual subset, since it is a G δ set. Lemma 3.10. There is a C 1 dense subset of diffeomorphisms E in O, such that any diffeomorphism in E is ergodic.
Before give the proof of Lemma 3.10, we need give some definitions and notations. log d(f n (x), f n (y)) < 0} and the Pesin unstable manifold of x, W u (x), is the Pesin stable manifold for f −1 . In the context, we use W s (resp. W u ) to denote the strong stable (resp. unstable) foliation.
As a corollary of Lemma 3.11 and Corollary 3.12, we have the following proposition: Proposition 3.13. Let f be a volume preserving diffeomorphism. Suppose f admits a dominated splitting E cs ⊕ E c 1 ⊕ E cu with dim(E c 1 ) = 1 and the set Λ s = {x; λ c (x) < 0} has positive volume. Then Lebesgue almost every point x ∈ Λ s has a local stable manifold W s loc (x) with dimension E cs ⊕ E c 1 . Moreover, W s loc (x) is tangent to the bundle E cs ⊕ E c 1 (this means, then the tangent space of W s loc (x) is contained in a small cone around E cs ⊕ E c 1 ).
Remark 3.14. Suppose Λ admits a partially hyperbolic splitting E s ⊕ E cs ⊕ E c ⊕ E u and there is a local strong stable leaf W s loc (x) tangent to E s (x) at some point x. If the center Lyapunov exponent at x is negative(resp. positive), then W s loc (x) ⊂ W s ε (x)(resp. W u loc (x) ⊂ W u ε (x)).

• Blenders
Topologically blenders are Cantor sets with distinctive geometric feature which is introduced by C. Bonatti and L.J.Diaz in [13] to give a slightly different mechanism for constructing non-Axiom A diffeomorphisms and robust heterodimensional cycles. Later, the notion of blender was motivated by L.J.Diaz in [20] and developed by F.Herz-M.A.Hertz-A.Tahzibi-R.Ures in [23] to produce a local source of stable ergodicity.
Definition 3.15. Let q, p be hyperbolic periodic points of a diffeomorphism f with stable index i and i + 1 respectively.
We say that f has a cs−blender of index i associated to (q, p) if there is a C 1 neighborhood U of f such that, for every g ∈ U, one has W s (p g ) is contained in the closure of W s (q g ), where q g , p g are the analytic continuation of q and p for g.
Define a cu−blender in an analogous way, by concerning f −1 .
There are several definitions of blender given in [13], [16], [20] and [24]. Our definition comes from the Definition and Lemma 1.10 in page 369 of [13]. And the proposition bellow which partially comes from the main result of [24] holds for our definition.  (q, p). Then there is a C 1 neighborhood U of f , such that for any g ∈ U, and for every i + 1−dimension disk D which is tangent to a cone around the bundle E cu ⊕ E c 1 , we have that where q g , p g are the analytic continuation of q and p for g.

Proof.
We only need to notice that W s (q) is tangent to the bundle E cs . From the definition of cs−blender, one can conclude the proof. It is remarkable that the distinctive blender property is a robust property. And homoclinical relation is also an open property. Hence, if a diffeomorphism f has a chain of blenders of index (i 0 , · · · , i 1 ), there is a neighborhood U of f , such that any g ∈ U has a chain of blenders of index (i 0 , · · · , i 1 ) as well.
Connecting lemma was proved by Hayashi [22] at first, and then was extended to the conservative setting by Wen and Xia [37,35](see also M.-C. Arnaud [2]). Later, from the proof of Hayashi's Connecting Lemma, Bonatti and Crovisier [12] extract a slightly stronger statement in the next lemma which permits to perform dynamically perturbations and create intersections between stable and unstable bundles.
Lemma 3.18 (Theorem 2.1(Connecting lemma) in [12], Theorem 3.10 in [24]). Let p, q be hyperbolic periodic points of a C r transitive diffeomorphism preserving a smooth measure m. Then, there exists a C 1 −perturbation g ∈ C r preserving m such that W s (p) ∩ W u (q) = ∅.  Proof: Take f ∈ O, then f has hyperbolic periodic points with index dim(E s ) and dim(E s )+1. By Proposition 3.16, there is an open set U 1 ⊂ O arbitrarily close to f , such that any g ∈ U 1 has a cs−blender of index dim(E s ) associated to (q 1,g , p 1,g ), where q 1,g and p 1,g denote the continuation of q 1 and p 1 for g. Note that O is an open set. Fix a diffeomorphism f 1 ∈ U 1 and by the same argument, we can find another open set U 2 ⊂ U 1 such that any g ∈ U 1 has a cs−blender of index dim(E s ) associated to (q 2,g , p 2,g ). Inductively, we obtain open sets U c(f ) ⊂ U c(f )−1 ⊂ · · · U 1 such that for any 1 ≤ i ≤ c(f ) and any g ∈ U i , g has a cs−blender of index dim(E s ) + i − 1 associated to (q i,g , p i,g ).
Take g ∈ U c(f ) , by Lemma 3.2 and using Lemma 3.18 twice, we get a diffeomorphism g 1 ∈ U c(f ) which is arbitrarily close to g, such that p 1,g1 and q 2,g1 are homoclinically related to each other. Do such perturbation c(f ) − 1 times, we obtain a diffeomorphism g c(f )−1 ∈ U c(f ) such that for any 1 ≤ i ≤ c(f ) − 1, one has that p i,g c(f )−1 and q i+1,g c(f )−1 are homoclinically related.
Let Proposition 3.20. Suppose f admits a dominated splitting E s ⊕ E c1 ⊕ · · · ⊕ E cc ⊕ E u with dim(E ci ) = 1 for any 1 ≤ i ≤ c, and has a chain of blenders Proof: It suffices to show that, there is n > 0 such that We will only prove the first part, since the second part comes analogously.
For any cone field around E c,j+1 ⊕ · · · ⊕ E c,c(f ) ⊕ E u , note that Df preserves this cone field and contracts its area uniformly. Thus, if D is tangent to a bundle which is tangent to such a cone field, then so do f n (D).
We prove this by induction. Since D ⋔ W s (p c(f ) ) = ∅, there exists a submanifold D 0 in D with dimension dim(E u ) + 1, which is tangent to a cone field around E c,c(f ) ⊕ E u and satisfies D 0 ⋔ W s (p c(f ) ) = ∅. By Proposition 3.16, we have Thus by λ-lemma, there is an integer n 0 > 0, such that f n0 (D 0 ) ⋔ W s (p c(f )−1 ) = ∅, since q c(f ) and p c(f )−1 are homoclinically related.
Continue the argument and then we can complete the proof.
• New Criteria of ergodicity Let p be a saddle of a C 1+α diffeomorphism f and Orb(p) denote the orbit of p. We write The following proposition is a direct corollary of Theorem A of [23] which presents a new criteria of ergodicity. Then f is ergodic.
Together with the following result of Avila, we will prove Lemma 3.10 and then complete the proof of Theorem 1.3. Proof of Lemma 3.10 Recall that the set O 1 ⊂ O is given in Proposition 3.19. We will use Proposition 3.21 to prove that any C 1+α volume preserving diffeomorphism f ∈ O 1 is ergodic. Since C ∞ volume preserving diffeomorphisms are dense in the C 1 topology by Lemma 3.22, we then can conclude that ergodic diffeomorphisms are dense in O 1 .
Combining Proposition 3.9 and Proposition 3.19, we know that f ∈ O 1 satisfies the following properties.
At first, we consider two special cases: j(f ) = 0 and j(f ) = c(f ). One can show the ergodicity of f by a theorem of Burns-Dolgopyat-Pesin [18]. Lemma 3.23 (Theorem 4 in [18]). Let f be a C 1+α partially hyperbolic diffeomorphism preserving a smooth measure. Assume that f is accessible and has negative center Lyapunov exponents on a set of positive measure. Then f is stably ergodic.
Because log Df | E c,j(f ) (x) dω(x) < 0 and log Df | E c,j(f )+1 (x) dω(x) > 0, one has ω(Λ cs ) > 0, ω(Λ cu ) > 0. Moreover, for ω − a.e. x, note that the subbundle E c,j(f ) (x) is dominated by the other bundle E c,j(f )+1 (x) and hence at least one of the Lyapunov exponents λ c j(f ) (x) and λ c j(f )+1 (x) should be nonzero. Thus we have that Λ cs ∪ Λ cu has full volume. Recall that f has a chain of cs−blenders of index (dim(E s ), · · · , dim(E s ) + c(f ) − 1) by (G2 ′ ), denoted by {q i , p i } c(f ) i=1 . Then for i = j(f ), f has a cs−blender of index dim(E s ) + j(f ) − 1 associated to (q j(f ) , p j(f ) ). By Proposition 3.21, Lemma 3.10 is a corollary of the following facts: ω−almost every point of Λ cu belongs to B s (p j(f ) , f ) and ω−almost every point of Λ cs belongs to B u (p j(f ) , f ). We only prove the first part and the proof of the second part is similar.
By Proposition 3.13, ω−almost every point x in Λ cu has local unstable manifold W u loc (x) of dimension d − (dim(E cs ) + j(f )) and moreover, W u loc (x) is tangent to the bundle E c,j(f )+1 ⊕ · · · ⊕ E c,c(f ) ⊕ E u .
Note that f has a cs−blender of index dim(E s ) + c(f ) − 1 associated to (q c(f ) , p c(f ) ). Because f is accessible, the orbit of almost every point is dense in the ambient manifold M . We can assume that the point x is arbitrarily close to the periodic points p c(f ) of index dim(E s ) + c(f ). Thus the strong unstable leaf at x should intersect the stable manifold at p c(f ) transversely, i.e., W u (x) ⋔ W s (p c(f ) ) = ∅. Since x ∈ Λ cu , by Remark 3.14, we have that W u (x) ⊂ W u loc (x). Then W u loc (x) ⋔ W s (p c(f ) ) = ∅. Now it follows from Proposition 3.20 that W u loc (x) ⋔ W s (p j(f ) ) = ∅ by taking D = W u loc (x). Hence x ∈ B s (p j(f ) ) and we conclude the proof.
Proof of Theorem 1.3 Since ergodic diffeomorphisms form a G δ set, Lemma 3.10 implies that there is a generic subset E among the set of volume preserving diffeomophisms far from tangencies, such that any diffeomorphism f ∈ E is ergodic.
We take R = O ∩ E and, together with Proposition 3.9, conclude the proof.

The proof of Theorem 1.5
Before proving Theorem 1.5, we need the following lemma: Lemma 3.24. (Theorem 0.1 in [19]) Let f be C 2 , volume-preserving, partially hyperbolic and center bunched. If f is essentially accessible, then f is ergodic, and in fact has the Kolmogorov property.
Proof of Theorem 1.5: For ∀f ∈ CPH 1 ω (M ), by Lemma 2.3, there is a stably accessible perturbation f 1 ∈ CPH 1 ω (M ). Applying Lemma 3.22, we can obtain anther perturbation f 2 which is C 2 and accessible. Using Lemma 3.24, f 2 is ergodic. Hence the subset consisting of all the ergodic diffeomorphisms is C 1 dense in CPH 1 ω (M ). Since it is well-known that this subset is a G δ set, we denote its intersection with int(HT ) and the residual set determined in Lemma 3.1 by R as the residual set we need.
For any f ∈ R, in the finest dominated splitting, there are center bundle with dimension larger than 1 by Lemma 3.7. Then the center exponents along this bundle equal, for, otherwise, suppose the exponents are different, then by Lemma 3.1, for almost every point, along the orbit, there is a dominated splitting. Because f is ergodic, this dominated splitting can be extended on the whole manifold. This contradicts our assumption that this splitting is a finest dominated splitting.