ACCESSIBILITY OF PARTIALLY HYPERBOLIC ENDOMORPHISMS WITH 1 D CENTER-BUNDLES ∗

We prove that partially hyperbolic endomorphisms with one dimensional center-bundles and non-trivial unstable bundles are stably accessible. And there is residual subset R of partially hyperbolic volume preserving endomorphisms with one dimensional center-bundles such that every f ∈ R is stably accessible. In the end, we prove the accessibility of Gan’s example.

Is f stably ergodic?
Here, f ∈ Diff 2 m (M ) is said to be stably ergodic, if there exists C 1 neighborhood U f such that every g ∈ U ∩ Diff 2 m (M ) is ergodic. The above example is a partially hyperbolic endomorphism. For stable ergodicity of partially hyperbolic diffeomorphisms, there are many important progress in recent years [4,6,8]. It is mostly motivated by Pugh-Shub's famous stable ergodicity conjecture: Conjecture 1.1 (Pugh-Shub [11]). On any compact manifold, ergodicity holds for an open and dense set of C 2 volume preserving partially hyperbolic diffeomorphisms.
In the same paper, Pugh-Shub posed a programme of the conjecture: Conjecture 1.2 (Pugh-Shub). Accessibility holds for an open and dense subset of C 2 partially hyperbolic diffeomorphisms, volume preserving or not.

Conjecture 1.3 (Pugh-Shub).
A partially hyperbolic C 2 volume preserving diffeomorphism wth the essential accessibility property is ergodic.
For a partially hyperbolic endomorphism f , it is said to be accessible if any two points can be connected by a path consists of stable manifolds and unstable manifolds. In partially hyperbolic diffeomorphisms with one dimensional center, Didier [5] proves the openness of accessibility. Theorem 1.1 ( [5]). Accessibility is C 1 -open among partially hyperbolic diffeomorphisms with one dimensional center bundle.
In partially hyperbolic volume-preserving diffeomorphisms with one dimensional center bundle, Hertz-Hertz-Ures's [8] prove the density of accessibility and consequently solve the stable ergodicity conjecture. Theorem 1.2 ( [8]). Stable accessibility/ergodicity is C r -dense among volume preserving partially hyperbolic diffeomorphisms with one dimensional center bundle, for all r ≥ 1/r ≥ 2.
In much important works [3,4,6,8,12]on stable ergodicity conjecture, the accessibility of dynamics is the key tool to prove the ergodicity of dynamics.
In this paper, we mainly generalize Didier's [5] and Hertz-Hertz-Ures's [8] partial results to endomorphisms. In partially hyperbolic endomorphism with one dimensional center, we characteristic the openness and density of accessibility. In fact, we want to completely generalize Hertz-Hertz-Ures's work [8] to partially hyperbolic endomorphisms with 1-dimensional center-bundles. Now, we are preparing the work on how to prove ergodicity by the accessibility [7].

orbit spaces
We firstly recall the orbit spaces of endomorphisms. It is formerly said to be inverse limits Let f : M ← be a regular(det(Df ) = 0) endomorphism. The orbit spaceM of f is the set consists with every orbit of f : The left-shift mapf is naturally induced by f , which satisfying π 0f = f π 0 . Here, π 0 is the projection of 0-coordinate fromM to M . LetM be the universal covering of M , and π :M → M be the covering map. The corresponding lifting mapf satisfying πf = f π.
By the commutative diagram, one has that an orbit ofx inM induces an orbit of x in M . For simplicity, the induced orbit is still denoted byx. Then, we have this natural map The following lemma is easy to check.

partially hyperbolic endomorphisms
Now, we introduce the definition of partially hyperbolic endomorphisms, which is very similar with diffeomorphisms'.
Definition 2.1. f is said to be partially hyperbolic, if TM has an invariant subbundle splitting TM = E s ⊕ E c ⊕ E u satisfying that there exists l ∈ N such that for anyx ∈M , any triple of unit vectors θ s ∈ E s (x), θ c ∈ E c (x), and θ u ∈ E u (x), one has that Also, we introduce the definition of partially hyperbolic endomorphisms under orbit spaces. Formally, we give a bundle structure TM on the orbit spaceM : for It is not difficult to show that the above definition is equivalent with the following. Definition 2.2. f is said to be partially hyperbolic, if TM has an invariant subbundle splitting TM = E s ⊕ E c ⊕ E u satisfying that there exists l ∈ N such that for anyx ∈M , any triple of unit vectors θ s ∈ E s (x), θ c ∈ E c (x), and θ u ∈ E u (x), one has that It is not difficult to show that for any x ∈ M , E s (x) does not depends on the choice of orbits of x. But generally speaking, E c and E u both depends on the choice of orbits. For instance, Gan's example in our paper. It is standard argument to show that Lemma 2.2. For any partially hyperbolic endomorphisms f , it has the following properties: • The partially hyperbolic splitting is unique.
• The splitting has uniform transversality: the angles between E s , E c , and E u are uniformly bounded from zero.
• The splitting is continuous: E σ (x) depends continuously on the orbitx, σ = s, c, u.
• Partially hyperbolic is persistent: there exists open neighborhood U of f such that for any g ∈ U, g is partially hyperbolic.
The proof is essentially same with diffeomorphisms' [2, Appendix B], just by replacing points by orbits. So, we omit its' proof.
For partially hyperbolic systems, we introduce the strong stable and unstable manifolds on orbit spaceM , M , andM . For any orbitx ∈M , the local strong unstable manifolds W u δ (f ,x) in orbit spaceM is the set the local strong unstable manifolds W u δ (f,x) in M is the set For a pointx in the universalM of M , its' local strong unstable manifolds W u δ (f ,x) (in the universalM of M ) is the set By these definitions, we can see that Also, it is not difficult to deduce that for anyx ∈M ⊂M , Similarily, we can define local strong stable manifolds W s In the end, we recall the classic stable and unstable manifolds theory: are tangent with E s and E u respectively, and vary continuously with respect to the orbit or point.

the accessibility classes
The accessibility is a powerful tool to check the ergodicity.
Definition 2.3. For a partially hyperbolic endomorphisms f , the accessibility class A f (x) is the set consists with the points which have su-paths from these points to the point x. Here the su-path is a concatenation of finitely many sub-paths, each of which lies entirely in a local stable/unstable manifold.
Definition 2.4. For a partially hyperbolic endomorphisms f , f is said to be accessible if f has only one accessibility class.
Note that the strong stable manifolds don't depend on the choice of orbits. Then if a partially hyperbolic endomorphism has no unstable bundles, it can't be accessible. However, we can use the diverse orbits of the point for non-inverse systems to get the accessibility. So, dim E s might be zero.
Let PH 1,1 (M ) be the set of all non-inverse partially hyperbolic endomorphism on M with dim E c ≤ 1 and dim E u ≥ 1.
Let r ≥ 1, PH r,1 m (M ) be the set of all C r partially hyperbolic volume preserving endomorphism on M with dim E c ≤ 1 and dim E u ≥ 1.
And let A f (x, δ, l) is the set consists with the points which have su-orbits at lsteps with δ-length, from these to the point x. For any subset X ⊂ M , A f (X, δ, l) . = x∈X A f (x, δ, l).

Other notations
It is not difficult to show that there exists δ f such that for anyx ∈ B(x, δ), π is diffeomorphic on B(x, δ f ). Let δ = min{δ 1 , δ f }, here δ 1 is the number in theorem 2.1. Through the paper, let W c δ (f ,x) be a smooth curve tangent with E c (M ) and centered atx. For anyX ⊂M , let For simplicity, let In the end, we emphasize that

main results
Partially hyperbolic endomorphisms f is said to be stably accessible, if there is a C 1 -neighborhood U such that every g ∈ U is accessible.
The next result characteristics the density of accessibility in PH r,1 m (M ). Theorem 3.2. There is a residual subset R of PH r,1 m (M ) such that for any f ∈ R, f is accessible, r > 1.
For Gan's example, We can prove that it is stably accessible.
Then, f is stably accessible.

the structure of accessibility classes
In the section, we focus endomorphisms in PH 1,1 (M ). The following is a basic and important lemma in the characteristic of accessibility classes. It happens in a small neighborhood. So its' proof is the same with diffeomorphisms' [8,Proposition A.4.].
be a accessibility class of f , the following conditions are equivalent: , the unstable space of y does not depend on the choice of orbits.
The first characteristic is paralled with the case of diffeomorphisms [8]. Proof. 1. Note that ϕM is dense inM , and the continuity of stable/unstalbe manifolds. Then "⇐=" is obvious. Now we deduce the left half. Let ε satisfying that for any W cu δ (f,x) and any y ∈ B(x, 2ε), W cu δ (f,x) ∩ W s δ (f, y) = ∅. On the contrary, we suppose there exists x ∈ A f (a), y ∈ W u ε (f,x), z ∈ W s ε (f, x), and an orbitẑ of z such that, W s δ (f, y)∩W u δ (f,ẑ ) = ∅. Take a curve W c δ (f,ẑ ) tangent with E c (f ), and x)). Since the stable manifolds vary continuously, γ is a continuous curve. Note that {z, w} ⊂ γ ⊂ A f (x). Then, by the continuity of unstable manifolds, It is not difficult to be deduced by the above item. 3. By the first characteristic of accessibility, we can see that A f (a) ⊂ T(f ) iff there exists ε such that, for any x ∈ A f (a), and any two orbitsx andx, W u ε (f,x ) ⊂ W u δ (f,x). Then, for A f (a) ⊂ T(f ), we have that for any x ∈ A f (a), the unstable space of x does not depend on the choice of orbits.
On the other hand, suppose that for any x ∈ A f (a), the unstable space of x does not depend on the choice of orbits. Then for any x ∈ A f (a) and any orbit {x i } of x, the unstable space of x i also does not depend on the choice of orbits, i < 0. On the contrary, we suppose that there exists x ∈ A f (a) and two orbitŝ x andx, W u ε (f,x ) ⊂ W u δ (f,x). Let ϕ i : B(x i , δ) → M satisfying f ϕ i = id and ϕ i (x i ) = x i−1 . Take a non-trivial E c -curve I c = [y, z] satisfying y ∈ W u ε (f,x ) and z ∈ W u δ (f,x). Take the corresponding triangle ∆(x, y, z), which consists with the segment of W u δ (f,x), the segment of W u δ (f,x) , and the segment I c = [y, z]. By the contracting on the segment of W u δ (f,x) and the segment of W u δ (f,x) , Inductively, we have that ϕ −n · · · ϕ −1 (∆(x, y, z)) ⊂ B(x −n , δ), n > 0.
By this contradiction, we get that if for any y ∈ A f (a), the unstable space of y does not depend on the choice of orbits, then A a (f ) ⊂ T(f ).

Remark 4.1.
• Obviously the condition in the second argument is robust. We believe that it is also necessary.
• For the case dim E s > 0, the latter condition in the third argument is not necessary. For example, f × Id, here f is non-special Anosov endomorphism, and Id is identity.
For anyx, let the stable manifold W s (f ,x) ofx is the set n<0f n (W s δ (f ,f −nx )).
The following is a directed consequence of the above lemma.  By the continuity of stable/unstable manifolds and the dominated splitting, we can see that the phenonema of 4-legs in the proof of above lemma is robust. Then by the lemma4.2, we get the following corollary.

C r openness and density of accessibility
Based on the characteristic of accessibility classes in above section, we prove the openness and density of accessibility. At first, we give its' openness.
Theorem 5.1. If f is accessible, then f is stable accessibility.
Under the above charateristic of accessibility classes, the proof is exactly same with diffeomorphisms' [5]. For completion, we give the proof. Among these systems, the accessibility is also a C r -density property.
Theorem 5.2. There is a residual subset R of PH r,1 m (M ) such that for any f ∈ R, f is accessible, r ≥ 1.
From now on, we focus endomorphism f ∈ PH 1,1 m (M ). By the volume-preserving, it is not difficult to see that |det(Df x )| ≥ 1 for any x ∈ M . Then for the case dim E u = 0, one has that E c is uniformly expanding, i.e., f is Anosov. For Anosov volume-preserving endomorphisms, they are born accessible and ergodic [1,13,14]. So, we suppose dim E u > 0. Then, we should use the diverse orbits of the point for non-inverse systems.
To prove the density, we modify the strategy in [8]: • Firstly prove that there is no periodical point in T f by a perturbation lemma. In our non-diffeomorphic case, the perturbation is totally different.
• In our non-diffeomorphic case, T(f ) is not totally invariant subset, we find another totally invariant subset T * (f ) ⊆ T(f ) replacing it. Then analyze on the boundary of T * (f ) to find an invariant accessible class W us (x); and by the hyperbolicity on W us (x), we find a periodical point. The analysis on the boundary of T(f ) is a local argument in [8]. In our non-diffeomorphic case, we projective the local center/stable/unstable manifold structure off into (M, f ), and follow the idea of [8] to analyze the dynamics in a small neighborhood in M.
We first give the key lemma.
Lemma 5.1. There is residual subset R of PH r,1 m (M ) such that for any f ∈ R and any periodical point The following is its perturbation form. m (M ) such that f ∈ U and for any g ∈ U, A g (p g ) is open. Proof. In [8], they deal with the case of diffeomorphisms. For the case of nondiffeomorphism, our strategy is to make perturbation in arbitraily small neighborhood of the periodic orbit such that, it preserves this periodic orbit and there is another orbit having different unstable space from the periodic orbit's.
If .
Obviously g preserve the orbitsp andp of f . Since p i ∈ B(p −1 , ε), i = −1, E su (f,p) = E su (g,p), and E u (f,f −1 (p)) = E u (g,f −1 (p)). By the derivative Dg on the point p −1 , we have that Then, E s (g, p) + E u (g,p) + E u (g,p) = T p M . By Lemma 4.2, one has that A g (p) = A g (p g ) is open. The proof of Lemma 5.1 is a common generic argument. For completion, we give the proof. Note that Ψ is lowercontinuous. By the classic semi-continuous theorem and Kupa-Smale theorem, we see that there is residual subset R 1 of PH r,1 m (M ) such that for any f ∈ R 1 , f is continuity point of Ψ and Per k (f ) is hyperbolic.
Take a dense subset {f n } n≥1 of R 1 . Suppose Per k (f n ) = {p 1 , · · · , p l }. Then, by the above claim and Corollary 4.2, we see that there is an open subset U n,k,1 such that f n ∈ U n,k,1 and for any g ∈ U n,k,1 , A g (p 1 g ) is open. And by Corollary 4.2 and the above claim again, we deduce that there is an open subset U n,k,2 ⊂ U n,k,1 such that f n ∈ U n,k,2 and for any g ∈ U n,k,2 , A g (p 2 g ) is open. Inductively, we have that there exists an open subset U n,k,l such that f n ∈ U n,k,l and for any g ∈ U n,k,l , Let U n,k = U n,k,l . By the continuity of Ψ in f , we see that for any g ∈ U n,k and any periodical points p g ∈ Per k (g), Since f is local diffeomorphism, intf −1 (A(x)) = ∅ for any x ⊂ T(f ). By Lemma 4.1, f −1 (T(f )) ⊆ T(f ) and f −1 (T(f )) is union of accessibility classes. By the induction, it is not difficult to show that for any positive integer n, f −n (T(f )) ⊆ T(f ) and f −n (T(f )) is union of accessibility classes. Then, it is not difficult to deduce that, • f is both negative invariant: f −1 (T * (f )) = T * (f ), and positive invariant: By the key lemma 5.1, there exists residual subset R of PH r,1 m (M ) such that for any f ∈ R, T(f ) ∩ Per(f ) = ∅. To the contrary, we suppose that there exists f ∈ R such that T(f ) = M . Since f is surjection, T * (f ) = ∅. Then, it is not difficult to take a closed curve (â,b) =Î ⊂ W c η (f ,b) and let I = (a, b) = π(Î) satisfying that a ∈ T * (f ), (a, b] ⊂ U * (f ), and η < δ 2 .
Case 2: there exists z ∈ (x, y) such that f k z ∈ W us δ (f,â). Since (x, y) ⊂ U * (f ), it contradicts the negative invariance of T * (f ). So, f k y ∈ W us δ (f,â). Duing to the hyperbolicity on W us (f,â), we have the following shadowing property: There exists ε such that if f n (x) ∈ W us ε (f,x) ⊂ T (f ), n > 0, then there exists a periodical point p ∈ W us δ (f,x). By the above claim, there is a periodical point contained in T(f ), which contradicts with T(f ) ∩ Per(f ) = ∅.
For the general shadowing property of endomorphisms, Bowen's argument should be still effective on the manifold W us . For completion, we give a trick proof of the above claim.
Proof of Claim 5.1. For simplicity, for any z ∈ T (f ), W σ (f,ẑ) is denoted by W σ (z), σ = us, u, s. Suppose x is not periodical. By the uniform expanding on the unstable manifold, we can take ε small enough such that n is big enough and satisfying the following properties: • For any z ∈ M , W u δ (z) ⊂ f n (W u δ 2 (f n z)).
• Let ϕ : W u δ (f n x) → W u δ (x) satisfying f n ϕ = id. for any y ∈ W u δ (f n x), W s δ (ϕ(y)) have exactly one point with W u δ (f n y). Then, we define a continuous endomorphism h on W u δ (f n x): h(y) = W s δ (ϕ(y)) ∩ W u δ (f n x).
Then, by the classic Brouwer fixed-point theorem , h has a fixed point y, i.e, ϕ(y) ∈ W s δ (y). By the uniform contraction and the coherence of stable manifolds, we see that {f mn (y)} m>0 ⊂ W s loc (y) is a Cauchy sequence. Obviously lim m>1 f mn (y) = p ∈ W us δ (x) be a periodical point of f . For the case dim E s = 0, we can get the periodical point of f by the last deduction.
Then, f is stably accessible.