The local $C^1$-density of stable ergodicity

The center bundle of a conservative partially hyperbolic diffeomorphism $f$ is called robustly non-hyperbolic if any conservative diffeomorphism which is $C^1$-close to $f$ has non-hyperbolic center bundle. In this paper, we prove that stable ergodicity is $C^1$-dense among conservative partially hyperbolic systems with robust non-hyperbolic center.


Introduction
Let M be a smooth compact, connected and boundless Riemannian manifold with dimension d ≥ 3, and µ be a smooth volume measure on M with µ(M ) = 1. Denote by Diff r µ (M ) the set of C r µ-preserving diffeomorphisms of M endowed with C r topology for r ≥ 1. If f ∈ Diff r µ (M ), we also call f is a conservative system. A diffeomorphism f : M → M is said to be partially hyperbolic, if f admits a nontrivial Df -invariant splitting of the tangent bundle T M = E s ⊕ E c ⊕ E u and numbers 0 < α s < α ′ c ≤ α ′′ c < α u such that α s < 1 < α u and for any x ∈ M , we have where m(Df | E ) is the minimum norm of Df | E , i.e., The subbundles E u , E c and E s are called unstable, center and stable bundle. Set β = dim(E β ) for β = s, c, u. Partial hyperbolicity is a robust property. That is to say, for any given partially hyperbolic diffeomorphism f of M , there is a C 1 neighborhood U of f in Diff 1 (M ) such that any g ∈ U is partially hyperbolic. We denote by PH r µ (M ) the family of C r conservative partially hyperbolic diffeomorphisms of M endowed with C r topology for r ≥ 1. Given f ∈ PH 1 µ (M ), the center bundle E c f of f is called robustly non-hyperbolic if there is a C 1 neighborhood U of f in PH 1 µ (M ) such that each g ∈ U has two ergodic measures µ 1 and µ 2 satisfy λ + µ 1 ≤ 0 and λ − µ 2 ≥ 0, where λ + µ 1 and λ − µ 2 are the largest and smallest Lyapunov exponents of µ 1 and µ 2 in E c g . We set : E c f is robustly non-hyperbolic}. Then P is a non-empty open subset of PH 1 µ (M ). The openness is obvious by the definition. On the other hand, if a conservative partially hyperbolic system f have two hyperbolic periodic points with indices s and s + c respectively, then f ∈ P. This implies that P is non-empty.
The main result of this paper is Theorem A. There is a subset D of P such that D is C 1 -dense in P and each f ∈ D is stable ergodic.
The study of stable ergodicity has a long-time history. In [2,3], by using Hopf Argument ( [22]), D. Anosov and J. Sinai established ergodicity of all C 2 volume-preserving uniformly hyperbolic systems (Anosov systems), including geodesic flows for compact manifolds of negative sectional curvature. In 1994, M. Grayson, C. Pugh and M. Shub ( [19]) gave the first nonuniformly hyperbolic example of a stably ergodic system. These systems are partially hyperbolic. Following this direction, Pugh and Shub believe that a little hyperbolicity goes a long way in guaranteeing ergodicity and,in [26,25], they posed the following Stable Ergodicity Conjecture: Conjecture. Stable ergodicity is C r -dense among conservative partially hyperbolic diffeomorphisms.
At the same time, Pugh and Shub gave a program to deal with this conjecture: they conjectured that stable accessibility is dense and essential accessibility implies ergodicity among volume-preserving, partially hyperbolic diffeomorphisms. In recent years, many advances have been made for this conjecture (e.g. see the survey [16,27]). For example, F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures ( [28]) proved that stable ergodicity is C ∞ -dense among partially hyperbolic diffeomorphisms with one-dimensional center bundle; K. Burns and A. Wilkinson ([17]) proved that essential accessibility implies ergodicity if the system is center bunched, and C. Bonatti, C. Matheus, M. Viana, and A. Wilkinson ([13]) proved the conjecture in the C 1 topology for one-dimensional center bundle.
As pointed in [31,32], many arguments of previous works (such as [17] and [28]) seem to be hard to generalize and have reached their limits in these directions. Recently, a new alternate criterion to establish ergodicity be obtained by F. Rodriguez Hertz, M. Rodriguez Hertz, A. Tahzibi and R. Ures in [29,31]. Using this argument, the authors proved the Pugh and Shub's Conjecture with two-dimensional center bundle in C 1 -topology.
Highly motivated by the Stably Ergodic Conjecture, our main result (Theorem A) of this paper provides a large class of conservative partially hyperbolic diffeomorphisms which can be C 1 approximated by stably ergodic systems. Unlike [17] or [31], these systems considered here are more general and the center dimension is not necessarily two.

Preliminaries
Given f ∈ Diff 1 µ (M ), by Oseledec Theory ( [24]), there is a µ-full invariant set O ⊂ M such that for every x ∈ O there exist a splitting (which is called Osledec splitting) and real numbers (the Lypunov exponents of µ) for every v ∈ E j (x) \ {0} and j = 1, 2, · · · , k(x). In the following, by counting multiplicity, we also rewrite the Lyapunov exponents of µ as It is obvious that the continuous points of the Lyapunov map , the distributions E u and E s are integrable and their integrable manifolds form two transversal foliations of M , the strongly stable and strongly unstable foliations of M , which we denote by W u and W s respectively. For every x ∈ M the leaves W u (x) and W s (x) of the foliations containing x are smooth immersed submanifolds in M called the strong unstable and strong stable global manifolds at x (see e.g. [14,21]).
Two points x, y ∈ M are called accessible if there are points x = z 0 , z 1 , · · · , z l−1 , z l = y, z i ∈ M such that z i ∈ W u (z i−1 ) or z i ∈ W s (z i−1 ) for i = 1, · · · , l. A diffeomorphism f is called an accessible diffeomorphism if it has the accessibility property, i.e., any pare points Accessibility is important to show the ergodicity of partially hyperbolic diffeomorphisms. In [18], D. Dolgopyat and A. Wilkinson proved that stable accessibility is C 1 dense. That is The following lemma can be find in [15].
and f is accessible. Then almost every orbit is dense in M .
An ergodic measure ν of f ∈ Diff r (M ) is called hyperbolic if all the Lyapunov exponents of ν is not zero. If r > 1, for every point x ∈ O, there are Pesin's stable and unstable manifolds which we denote by W s (x) and W u (x). If f is also partially hyperbolic, we Given a diffeomorphism f and a f -invariant set K ⊂ M , a Df -invariant splitting Dominated splitting is unique, transverse and continuous. Moreover, the dominated splitting has some robust properties (see e.g. [12]). A dominated splitting E 1 ⊕ · · · ⊕ E k is called the finest dominated splitting if there is no dominated splitting in each invariant bundle E i for all i = 1, · · · , k. Moreover, a splitting is called the robust finest dominated splitting if the continuation of the splitting is the finest dominated splitting of the C 1 perturbation diffeomorphism.
For a hyperbolic periodic point P of f , we denote by ind(P ) the index of P , where the index of P refers to the dimension of the stable bundle of P . The homoclinic class of a hyperbolic saddle P of a diffeomorphism f , denoted by H(P, f ), is the closure of the transverse intersections of the invariant manifolds (stable and unstable ones) of the orbit of P . Connecting Lemma was firstly proved by S. Hayashi ([20]) and was extended by L. Wen and Z. Xia in [33] (see also [4]). The following Connecting Lemma which established by C. Bonatti and S. Crovisier from the proof of Hayashi's will permit us to create intersections between stable and unstable manifolds. [9]) Let Q, P be hyperbolic periodic points of a C r (r ≥ 1) transitive diffeomorphism preserving a smooth measure µ. Then, there exists a C 1 -perturbation g ∈ C r preserving µ such that W s (P ) ∩ W u (Q) = ∅.

Lemma 2.4. (Connecting Lemma
Blender has been introduced firstly in [10], and it is a very useful tool to understand the dynamical properties such as the transitivity and ergodicity (e.g. see [10,11,30,31]). There are several definitions of blender( [10,12,23,30]). The following definition comes from [11] and [30]. Definition 2.5. Let P, Q be hyperbolic periodic points of a diffeomorphism f with index i and i + 1 respectively. We say that f has a cu-blender of index i associated to (P, Q) if (1) P is a partially hyperbolic periodic point of f such that Df is expending on E c and dim(E c ) = 1; (2) there is a small open set Bl u (P ) such that every (d − i)-strip well placed in Bl u (P ) transversely intersects W s (P ); (3) W u (Q) ∩ Bl u (P ) contains a vertical disk D through Bl u (P ), i.e., D is a (d − i − 1)disk which is centered at a point in Bl u (P ), the radius of D is much bigger than the radius of Bl u (P ) and D is almost tangent to E u ; (4) this property is C 1 -robust. Define a cs-blender in an analogous way by concerning f −1 .
We also will use the following two lemmas.

Proof of Theorem 1
We first give a lemma which is important to the proof of Theorem A. Proof. This is the direct result of Lemma 2.3 and the conservative version of Theorem 1.1 of [1](or see the Lemma 3.8 of [23]). ✷ Now, we recall the criteria of ergodicity of [31]. Given a diffeomorphism f and a hyperbolic periodic point P , we define two invariant sets: where O is the set of Oseledec regular points and W s (W u ) is the Pesin global stable (unstable) manifold. and f is ergodic on Λ(P ). Moreover, f is non-uniformly hyperbolic on Λ(P ).
In the following Lemma, we give a dense subset D ⊂ PH 1 µ (M ). In fact, to prove Theorem A, we only need to prove that the stable ergodicity can C 1 approximate to each system of a dense subset of P.
such that for any f ∈ D, we have (1) f is stably accessible and (2) there exists a robust finest dominated splitting of Df , such that the Lyapunov exponents at x in E i are equal for µ-a.e. x ∈ M and all i = 1, 2, · · · , k.
Proof. By Lemma 2.2 of [23], there is an open and dense subset D 1 ⊂ PH 1 µ (M ) such that, for each f ∈ D 1 , Df has a robust finest dominated splitting T M = E 1 ⊕ E 2 ⊕ · · · ⊕ E k . By Lemma 2.1 and 2.3, there is a residual set D 2 ⊂ D 1 ⊂ PH 1 µ (M ) such that each f ∈ D 2 is stably accessible and M is the unique homoclinic class. Set D = D 2 ∩ R 3 , where R 3 refers to Lemma 2.6. We shall prove that D satisfies the lemma.
For any 1 ≤ i < d and l ∈ N, denote D i (f, l) by the set of points x such that there is a l-dominated splitting of index i along the orbit of x ∈ M . Then D i (f, l) is a compact invariant set. Set Γ i (f, l), for i = 1, 2, · · · , d − 1.
We shall show that, up to zero measure, either Γ i (f, ∞) = M or D i (f, l) = M for some l. In fact, if µ(D i (f, l)) = 0 for all l, then Γ i (f, ∞) = M mod 0. If µ(D i (f, l)) > 0 for some l, by Lemma 2.6 and 2.7, for any η > 0 there is g ∈ Diff 2 µ (M ) which is C 1 close to f and a g-invariant Borel set Λ such that Λ ⊂ B η (D i (f, l)) and µ( Λ△D i (f, l)) < η. Since f is stably accessible and g is C 2 , Lemma 2.2 implies that Λ is dense in M . So, Noting that (3.1) is finest dominates splitting, D i (f, l) = M if and only if i = dim(E 1 )+ · · · + dim(E j ) for some j = 1, · · · , k. By [8], for µ-a.e. x ∈ M and any i = 1, 2, · · · , k, the Lyapunov exponents in E i are equal.
Proof of Theorem A. For any f ∈ PH 1 µ , r ≥ 1 and ε > 0, we set To prove Theorem A, we only need to prove that for any f ∈ D ∩ P ∩ R 0 and any ε > 0, there is g ∈ U 2 (f, ε) such that g is stably ergodic.
Since f ∈ P, there is an ergodic measure µ 1 such that λ + µ 1 ≤ 0, where λ + µ 1 is the largest and smallest Lyapunov exponent of µ 1 in E c f . By Ergodic Closing Lemma ( [4]), µ 1 -a.e. point is well closable. Then, for any ǫ 1 ≤ ε 3 , there are f ′ ∈ U 1 (f, ǫ 1 2 ) and a periodic point P ′ of f ′ such that λ s+c (P ′ ) ≤ ǫ 1 3 . If λ s+c (P ′ ) < 0, then P ′ is a hyperbolic periodic point with index s + c. Otherwise, using the conservative version of Frank's Lemma, one can get a new diffeomorphism f ′′ ∈ U 1 (f ′ , ǫ 1 2 ) which has the periodic point P ′′ with index s + c. Anyway, for the ǫ 1 , there is f 1 ∈ U 1 (f, ǫ 1 ) such that f 1 has a hyperbolic periodic point P 1 with index s + c.
Since f has a robustly non-hyperbolic center bundle, if we select ǫ 1 small enough, there is an ergodic measure ν of f 1 such that λ − ν ≥ 0, where λ − ν is the smallest Lyapunov exponent of ν in E c f 1 . By the similar discussion as above, for any ǫ 2 ≤ ǫ 1 , there is f 2 ∈ U 1 (f 1 , ǫ 2 ) such that f 2 has a hyperbolic periodic point Q with index s. Since P 1 is a hyperbolic periodic point of f 1 , the continuation P of P 1 is a hyperbolic point of f 2 and has the same index as the P 1 's if ǫ 2 is small enough.
That is to say, for any ǫ 1 > 0, there is f 2 ∈ U 1 (f, 2ǫ 1 ) such that f 2 has two hyperbolic periodic points P and Q with indices s + c and s respectively.
If ǫ 3 ∈ (0, ǫ 2 ) is small enough, any g ∈ U 1 (f 2 , ǫ 3 ) has two hyperbolic periodic points of indices s and s + c by the hyperbolicity of P and Q. Moreover, by Lemma 3.1, any g ∈ U 1 (f 2 , ǫ 3 ) ∩ R 4 has a dense subset of saddles of index i for all i ∈ {s, s + 1, · · · , s + c}. So, there is an open set V 0 ⊂ U 1 (f 2 , ǫ 3 ) such that each g ∈ V 0 has a subset of hyperbolic periodic points of index i for all i ∈ {s, s + 1, · · · , s + c}.
Lemma 3.5. If there are two partially hyperbolic points P ′ , Q ′ with indices i, i + 1 and one-dimension center, then there exists a cu-blender of index i associated to (P, Q) and P, Q are homoclinic related to P ′ , Q ′ respectively.
Proof. This is the conservative version of subsection 4.1 of [11]. One also can see the construction in [30]. Now we continue to prove the Theorem. Selecting a diffeomorphism g 0 ∈ V 0 , since g 0 has two saddles of indices s + c − 1 and s + c, by Lemma 3.5 and the robust property of blender, there is an open subset V 1 ⊂ V 0 , such that any g ∈ V 1 has a cu-blender of index s + c − 1. Selecting a diffeomorphism g 1 ∈ V 1 , since g 1 has two saddles of indices s + c − 2 and s + c − 1, by Lemma 3.5 and the robust property of blender again, there is an open subset V 2 ⊂ V 1 , such that any g ∈ V 2 has a cu-blender of index s + c − 2. Noting that V 2 ⊂ V 1 , g also has a cu-blender of index s + c − 1. Inductively, we obtain open sets V c ⊂ · · · ⊂ V 1 such that for any 1 ≤ i ≤ c and any g ∈ V i , g has i cu-blenders of indices s + c − 1, s + c − 2, · · · , s + c − i respectively. Especially, for any g ∈ V c and any s ≤ i < s + c, there is cu-blender of index i associated to (P i,g , Q i+1,g ).
To continue the proof, we need the following lemma.
Lemma 3.6. If ε is small enough, then there is g ∈ V c ∩ Diff 2 (M ) such that g is stably ergodic.