The ergodic shadowing property from the robust and generic view point

Abstract

In this paper, we discuss that if a diffeomorphisms has the $C^1$-stably ergodic shadowing property in a closed set, then it is a hyperbolic elementary set. Moreover, $C^1$-generically: if a diffeomorphism has the ergodic shadowing property in a locally maximal closed set, then it is a hyperbolic basic set.

MSC:34D30, 37C20.

1 Introduction

Let M be a closed $C^{\mathrm{\infty }}$ manifold, and let $Diff(M)$ be the space of diffeomorphisms of M endowed with the $C^1$-topology. Denote by d the distance on M induced from a Riemannian metric $\parallel \cdot \parallel$ on the tangent bundle TM. Let $f\in Diff(M)$. For $\delta >0$, a sequence of points ${\{x_i\}}_{i=a}^b$ ($-\mathrm{\infty }\le a<b\le \mathrm{\infty }$) in M is called a δ-pseudo orbit of f if $d(f(x_i),x_{i+1})<\delta$ for all $a\le i\le b-1$. For given $x,y\in M$, we write $x⇝y$ if for any $\delta >0$, there is a δ-pseudo orbit ${\{x_i\}}_{i=a}^b$ ($a<b$) of f such that $x_a=x$ and $x_b=y$. Let Λ be a closed f-invariant set. We say that f has the shadowing property in Λ if for every $ϵ>0$ there is $\delta >0$ such that, for any δ-pseudo orbit ${\{x_i\}}_{i=a}^b\subset \mathrm{\Lambda }$ of f ($-\mathrm{\infty }\le a<b\le \mathrm{\infty }$), there is a point $y\in \mathrm{\Lambda }$ such that $d(f^i(y),x_i)<ϵ$ for all $a\le i\le b-1$. If $\mathrm{\Lambda }=M$, then f has the shadowing property. The shadowing property usually plays an important role in the investigation of stability theory and ergodic theory. For instance, Sakai [1] proved that if f has the $C^1$-robustly shadowing property, then f is structurally stable. Now we introduce the notion of the ergodic shadowing property which was introduced and studied by [2]. Lee has shown in [3] that if f belongs to the $C^1$-interior of the set of all diffeomorphisms having the ergodic shadowing property, then it is structurally stable diffeomorphisms. In [4], Lee showed that if f is local star condition and has the ergodic shadowing property on the homoclinic class, then it is hyperbolic. For any $\delta >0$, a sequence $\xi ={\{x_i\}}_{i\in \mathbb{Z}}$ is a δ-ergodic pseudo orbit of f if for $N{p}_n^{+}(\xi ,f,\delta )=\{i:d(f(x_i),x_{i+1})\ge \delta \}\cap \{0,1,\dots ,n-1\}$, and $N{p}_n^{-}(\xi ,f,\delta )=\{-i:d(f^{-1}(x_{-i}),x_{-i-1})\ge \delta \}\cap \{-n+1,\dots ,-1,0\}$

$$\underset{n\to \mathrm{\infty }}{lim}\frac{\mathrm{\#}N{p}_n^{+}(\xi ,f,\delta )}{n}=0\text{and}\underset{n\to -\mathrm{\infty }}{lim}\frac{\mathrm{\#}N{p}_n^{-}(\xi ,f,\delta )}{n}=0.$$

Here #A is the number of elements of the set A. We say that f has the ergodic shadowing property in Λ (or $f{|}_{\mathrm{\Lambda }}$ has ergodic shadowing) if for any $ϵ>0$, there is a $\delta >0$ such that every δ-ergodic pseudo orbit $\xi ={\{x_i\}}_{i\in \mathbb{Z}}\subset \mathrm{\Lambda }$ of f there is a point $z\in \mathrm{\Lambda }$ such that, for $N{s}_n^{+}(\xi ,f,z,ϵ)=\{i:d(f^i(z),x_i)\ge ϵ\}\cap \{0,1,\dots ,n-1\}$, and $N{s}_n^{-}(\xi ,f,z,ϵ)=\{-i:d(f^{-i}(z),x_{-i})\ge ϵ\}\cap \{-n+1,\dots ,-1,0\}$,

$$\underset{n\to \mathrm{\infty }}{lim}\frac{\mathrm{\#}N{s}_n^{+}(\xi ,f,z,ϵ)}{n}=0\text{and}\underset{n\to -\mathrm{\infty }}{lim}\frac{\mathrm{\#}N{s}_n^{-}(\xi ,f,z,ϵ)}{n}=0.$$

Note that f has the ergodic shadowing property on Λ and f has the ergodic shadowing property in Λ are different notions. That is, the shadowing point is in M or Λ. In the first notion, the shadowing point is in M. In the second notion, the shadowing point is in Λ. In this paper we consider the latter case.

We say that Λ is locally maximal if there is a compact neighborhood U of Λ such that

$$\bigcap _{n\in \mathbb{Z}}{f}^n(U)={\mathrm{\Lambda }}_f(U)=\mathrm{\Lambda }.$$

Now, we introduce the notion of the $C^1$-stably ergodic shadowing property in a closed set.

Definition 1.1 Let Λ be a closed f-invariant set. We say that f has the $C^1$-stably ergodic shadowing property in Λ if

• (i) there is a neighborhood U of Λ and a $C^1$-neighborhood $\mathcal{U}(f)$ of f such that ${\mathrm{\Lambda }}_f(U)=\mathrm{\Lambda }={\bigcap }_{n\in \mathbb{Z}}{f}^n(U)$ (that is, Λ is locally maximal);

• (ii) for any $g\in \mathcal{U}(f)$, g has the ergodic shadowing property on ${\mathrm{\Lambda }}_g(U)={\bigcap }_{n\in \mathbb{Z}}{g}^n(U)$, where ${\mathrm{\Lambda }}_g(U)$ is the continuation of Λ.

We say that Λ is hyperbolic if the tangent bundle $T_{\mathrm{\Lambda }}M$ has a Df-invariant splitting $E^s\oplus E^u$ and there exist constants $C>0$ and $0<\lambda <1$ such that

$$\parallel D_x{f}^n{|}_{E_x^s}\parallel \le C{\lambda }^n\text{and}\parallel D_x{f}^{-n}{|}_{E_x^u}\parallel \le C{\lambda }^n$$

for all $x\in \mathrm{\Lambda }$ and $n\ge 0$. If $\mathrm{\Lambda }=M$, then f is Anosov. We say that Λ is a basic set (resp. elementary set) if $f{|}_{\mathrm{\Lambda }}$ is transitive (resp. mixing) and locally maximal. Note that if Λ is hyperbolic, then we can easily show that there is a periodic point such that the orbit of the periodic point is dense in the set. Then we get the following.

Theorem 1.2 [[5], Theorem 3.3]

Let Λ be a closed f-invariant set. If f has the $C^1$-stably ergodic shadowing property in Λ, then it is a hyperbolic elementary set.

Corollary 1.3 If f belongs to the $C^1$-interior of the set of all diffeomorphisms having the ergodic shadowing property, then it is transitive Anosov.

We say that a subset $\mathcal{G}\subset Diff(M)$ is residual if contains the intersection of a countable family of open and dense subsets of $Diff(M)$; in this case is dense in $Diff(M)$. A property P is said to be $C^1$-generic if P holds for all diffeomorphisms which belong to some residual subset of $Diff(M)$. We use the terminology ‘for $C^1$-generic f’ to express ‘there is a residual subset $\mathcal{G}\subset Diff(M)$ such that, for any $f\in \mathcal{G}\dots$ .’ In [6], Abdenur and Díaz proved that if tame diffeomorphisms has the shadowing property, then it is hyperbolic. Still open is the question if $C^1$-generically: f is shadowable, then is it hyperbolic?

Recently, Ahn et al. [7] have given a partial answer which is $C^1$-generically: if a locally maximal homoclinic class is shadowing, then it is hyperbolic. Lee has shown in [8] that $C^1$-generically: if f has the limit shadowing property on the homoclinic class, then it is hyperbolic. Inspired by this, we consider that $C^1$-generically: f has the ergodic shadowing property in a locally maximal closed set. Then we have the following.

Theorem 1.4 For $C^1$-generic f, if f has the ergodic shadowing property in a locally maximal closed set Λ, then it is a hyperbolic elementary set. Moreover, $C^1$-generically: if f has the ergodic shadowing property, then it is transitive Anosov.

2 Proof of Theorem 1.4

Let $P(f)$ be the set of periodic points of f. If $f{|}_{\mathrm{\Lambda }}$ is transitive, then every $p\in \mathrm{\Lambda }\cap P(f)$ is saddle, that is, there is no eigenvalues of $D_p{f}^{\pi (p)}$ with modulus equal to 1, at least one of them is greater than 1, at least one of them is smaller than 1, where $\pi (p)$ is the minimum period of p.

Lemma 2.1 [[2], Corollary 3.5]

If f has the ergodic shadowing property in Λ, then $f{|}_{\mathrm{\Lambda }}$ is mixing.

By Lemma 2.1, f has the ergodic shadowing property in Λ, then $f{|}_{\mathrm{\Lambda }}$ is mixing, and so $f{|}_{\mathrm{\Lambda }}$ is transitive. Thus $p\in \mathrm{\Lambda }\cap P(f)$ is neither a sink nor a source.

Lemma 2.2 [[2], Lemma 3.2]

If f has the ergodic shadowing property in Λ, then f has a finite shadowing property in Λ.

We say that f has the finite shadowing property on Λ if for any $ϵ>0$ there is $\delta >0$ such that, for any finite δ-pseudo orbit $\{x_0,x_1,\dots ,x_n\}\subset \mathrm{\Lambda }$, there is $y\in M$ such that $d(f^i(y),x_i)<ϵ$ for all $0\le i<n$. In [[9], Lemma 1.1.1], Pilyugin showed that f has a finite shadowing shadowing property on Λ, then f has the shadowing property on Λ.

Lemma 2.3 Let f have the ergodic shadowing in Λ and Λ be locally maximal in U. Then the shadowing point taken from Λ.

Proof Let f have the ergodic shadowing property in Λ, and let U be a locally maximal of Λ. For any $ϵ>0$, let $\delta >0$ be the number of the ergodic shadowing property of f. Take a sequence $\gamma ={\{x_i\}}_{i=0}^n$ ($n\ge 1$) such that γ is a δ-pseudo orbit of f and $\gamma \subset \mathrm{\Lambda }$. As in the proof of [[2], Lemma 3.1], there is a δ-pseudo orbit $\eta ={\{x_i\}}_{i=n}^0$ such that $\eta \subset \mathrm{\Lambda }$. Then we set $\xi =\{\dots ,\gamma ,\eta ,\gamma ,\eta ,\dots \}$ is a δ-ergodic pseudo orbit of f. Clear that $\xi \subset \mathrm{\Lambda }$. Since f has the ergodic shadowing property in Λ, ξ can be ergodic shadowed by some point $y\in \mathrm{\Lambda }$. By Lemma 2.2, there is $\gamma \in \xi$ such that $d(f^i(y),x_i)<ϵ$ for $0\le i\le n-1$. By [[9], Lemma 1.1.1], f has the shadowing property on Λ. Since Λ is locally maximal in U, the shadowing point $y\in \mathrm{\Lambda }$. □

Let $p\in P(f)$ be a hyperbolic saddle with period $\pi (p)>0$. Then there are the local stable manifold $W_{ϵ}^s(p)$ and the local unstable manifold $W_{ϵ}^u(p)$ of p for some $ϵ=ϵ(p)>0$. It is easily seen that if $d(f^n(x),f^n(p))\le ϵ$ for all $n\ge 0$, then $x\in W_{ϵ}^s(p)$, and if $d(f^n(x),f^n(p))\le ϵ$ for all $n\le 0$, then $x\in W_{ϵ}^u(p)$. The stable manifold $W^s(p)$ and the unstable manifold $W^u(p)$ defined as following. It is well known that if p is a hyperbolic periodic point of f with period k, then the sets

$$\begin{array}{c}{W}^s(p)=\{x\in M:f^{kn}(x)\to p\text{ as }n\to \mathrm{\infty }\}\text{and}\hfill \\ W^u(p)=\{x\in M:f^{-kn}(x)\to p\text{ as }n\to \mathrm{\infty }\}\hfill \end{array}$$

are $C^1$-injectively immersed submanifolds of M.

Lemma 2.4 Let $p,q\in P(f)$ be hyperbolic saddles. If f has the ergodic shadowing property in a closed set Λ, then $W^s(p)\cap W^u(q)\ne \mathrm{\varnothing }$, and $W^u(p)\cap W^s(q)\ne \mathrm{\varnothing }$.

Proof Let $p,q\in P(f)$ be hyperbolic saddles, and let U be a locally maximal neighborhood of Λ. Suppose that f has the ergodic shadowing property in a locally maximal Λ. Since p and q are hyperbolic, there are $ϵ(p)>0$ and $ϵ(q)>0$ as in the above. Take $ϵ=min\{ϵ(p),ϵ(q)\}/4$ and let $0<\delta \le ϵ$ be the number of the ergodic shadowing property of f. For simplicity, we may assume that $f(p)=p$ and $f(q)=q$. Since f has the ergodic shadowing property in Λ, $f{|}_{\mathrm{\Lambda }}$ is chain transitive. Then we can construct a finite δ-pseudo orbit form p to q as follows: $x_0=p$, $x_n=q$ ($n\ge 1$), and $d(f(x_i),x_{i+1})<\delta$ for all $0<i<n-1$. Put (i) $x_{-i}=f^{-i}(p)$, for all $i\le 0$, and (ii) $x_{n+i}=f^i(q)$ for all $i\ge 0$. Then we have the sequence $\xi ={\{x_i\}}_{i\in \mathbb{Z}}=\{\dots ,p,x_1,x_2,\dots ,x_n,x_{n+1},\dots \}$. It is clearly a δ-ergodic pseudo orbit of f. Since f has the ergodic shadowing property in Λ and locally maximal, by Lemma 2.2, f has the finite shadowing property on Λ and so, by [[9], Lemma 1.1.1], f has the shadowing property in Λ. By the shadowing property in Λ, we can show that $Orb(y)\subset W^u(p)\cap W^s(q)$, and so $W^u(p)\cap W^s(q)\ne \mathrm{\varnothing }$. The other case is similar. □

A diffeomorphism f is Kupka-Smale if their periodic points of f are hyperbolic and if $p,q\in P(f)$, then $W^s(p)$ is transversal to $W^u(q)$. Then it is $C^1$-residual in $Diff(M)$. Denote by $\mathcal{KS}(M)$ the set of all Kupka-Smale diffeomorphisms. The following was proved by [10].

Lemma 2.5 [[10], Lemma 2.4]

Let Λ be locally maximal in U, and let $\mathcal{U}(f)$ be given. If for any $g\in \mathcal{U}(f)$, $p\in {\mathrm{\Lambda }}_g(U)\cap P(g)$ is not hyperbolic, then there is $g_1\in \mathcal{U}(f)$ such that $g_1$ has two hyperbolic periodic points $p,q\in {\mathrm{\Lambda }}_{g_1}(U)$ with different indices.

Denote by $\mathcal{F}(M)$ the set of $f\in Diff(M)$ such that there is a $C^1$ neighborhood $\mathcal{U}(f)$ of f such that, for any $g\in \mathcal{U}(f)$, every $p\in P(g)$ is hyperbolic. In [11], Hayashi proved that $f\in \mathcal{F}(M)$ if and only if f satisfies both Axiom A and the no-cycle condition. We say that f is the local star condition diffeomorphism if there exist a $C^1$-neighborhood $\mathcal{U}(f)$ and a neighborhood U of Λ such that, for any $g\in \mathcal{U}(f)$, every $p\in {\mathrm{\Lambda }}_g(U)\cap P(g)$ is hyperbolic (see [12]). Denote by $\mathcal{F}(\mathrm{\Lambda })$ the set of all local star diffeomorphisms. Note that there are a $C^1$-neighborhood $\mathcal{U}(f)$ and a neighborhood U of p such that, for all $g\in \mathcal{U}(f)$, there is a unique hyperbolic periodic point $p_g\in U$ of g with the same period as p and $index(p_g)=index(p)$. Here $index(p)=dim{E}_p^s$, and the point $p_g$ is called the continuation of p.

Lemma 2.6 [[13], Lemma 2.2]

There is a residual set ${\mathcal{G}}_1\subset Diff(M)$ such that, for any $f\in {\mathcal{G}}_1$, if for any $C^1$-neighborhood $\mathcal{U}(f)$ of f, there exists $g\in \mathcal{U}(f)$ such that two hyperbolic periodic points $p_g,q_g\in P(g)$ with $index(p_g)\ne index(q_g)$, then f has two hyperbolic periodic points $p,q\in P(f)$ with $index(p)\ne index(q)$.

Lemma 2.7 There is a residual set ${\mathcal{G}}_2\subset Diff(M)$ such that, for any $f\in {\mathcal{G}}_2$, if f has the ergodic shadowing property in a locally maximal Λ, then for any $p,q\in \mathrm{\Lambda }\cap P(f)$

$$index(p)=index(q).$$

Proof Let $f\in {\mathcal{G}}_2={\mathcal{G}}_1\cap \mathcal{KS}(M)$, and let $p,q\in \mathrm{\Lambda }\cap P(f)$ be hyperbolic saddles. Suppose that f has the ergodic shadowing property in a locally maximal Λ. Then by Lemma 2.4 $W^s(p)\cap W^u(q)\ne \mathrm{\varnothing }$ and $W^u(p)\cap W^s(q)\ne \mathrm{\varnothing }$. Since $f\in {\mathcal{G}}_2$, $W^s(p)⋔W^u(q)\ne \mathrm{\varnothing }$ and $W^u(p)⋔W^s(q)\ne \mathrm{\varnothing }$. This means that $p\sim q$ and so $index(p)=index(q)$. □

Let p be a periodic point of f. For $0<\delta <1$, we say that p has a δ-weak eigenvalue if $D{f}^{\pi (p)}(p)$ has an eigenvalue λ such that ${(1-\delta )}^{\pi (p)}<|\lambda |<{(1+\delta )}^{\pi (p)}$. We say that a periodic point has a real spectrum if all of its eigenvalues are real and simple spectrum if all its eigenvalues have multiplicity one. Denote by $P_h(f)$ the set of all hyperbolic periodic points of f.

Lemma 2.8 [[14], Lemma 5.1]

There is a residual set ${\mathcal{G}}_3\subset Diff(M)$ such that, for any $f\in {\mathcal{G}}_3$:

• For any$\delta >0$, if for any$C^1$-neighborhood$\mathcal{U}(f)$of f there exist$g\in \mathcal{U}(f)$and$p_g\in P_h(g)$with a δ-weak eigenvalue, then there is$p\in P_h(f)$with a 2δ-weak eigenvalue.

• For any$\delta >0$, if$q\in P_h(f)$with a δ-weak eigenvalue and a real spectrum, then there is$p\in P_h(f)$with a δ-weak eigenvalue with a simple real spectrum.

Lemma 2.9 There is a residual set ${\mathcal{G}}_4\subset Diff(M)$ such that, for any $f\in {\mathcal{G}}_4$, if f has the ergodic shadowing property in a locally maximal Λ, then there exists $\eta >0$ such that, for any $q\in \mathrm{\Lambda }\cap P_h(f)$, q has no η-weak eigenvalues.

Proof Let $f\in {\mathcal{G}}_4={\mathcal{G}}_2\cap {\mathcal{G}}_3$ have the ergodic shadowing property in a locally maximal Λ. We will derive a contradiction. Suppose that, for any $\eta >0$, there is $q\in \mathrm{\Lambda }\cap P_h(f)$ such that q has an η-weak eigenvalue. By Franks’ lemma, there is g $C^1$-close to f such that p is not hyperbolic. By Franks’ lemma and Lemma 2.5, there is h $C^1$-nearby g and $C^1$-close to f such that h has tow hyperbolic periodic points $q_h$, ${\gamma }_h$ with different indices. Since $f\in {\mathcal{G}}_1$, and it is locally maximal, by Lemma 2.6 f has two hyperbolic periodic points q, γ in Λ. Since f has the ergodic shadowing property in Λ, this is a contradiction by Lemma 2.7. □

Proposition 2.10 There is a residual set ${\mathcal{G}}_4\subset Diff(M)$ such that, for any $f\in {\mathcal{G}}_4$, if f has the ergodic shadowing property in Λ, then $f\in \mathcal{F}(\mathrm{\Lambda })$.

Proof Let $f\in {\mathcal{G}}_4$ have the ergodic shadowing property in a locally maximal Λ. Suppose by contradiction that $f\notin \mathcal{F}(\mathrm{\Lambda })$. Then there are g $C^1$-close to f and $q\in P(g)$ such that q has a η-weak eigenvalue. Then by Lemma 2.9, we get a contradiction. Thus $f\in \mathcal{F}(\mathrm{\Lambda })$. □

Proposition 2.11 [[15], Proposition A]

Let $f\in {\mathcal{G}}_4$, and let Λ be locally maximal. If f has the ergodic shadowing property in Λ, then there are $m>0$, $C\ge 1$ and $\lambda \in (0,1)$ such that, for any $p\in \mathrm{\Lambda }\cap P(f)$ with $\pi (q)>m$, we have

$$\begin{array}{c}\prod _{i=0}^{\pi (p)-1}\parallel D{f}^m{|}_{E^s(f^{mi}(x))}\parallel \le C{\lambda }^{\pi (p)},\hfill \\ \prod _{i=0}^{\pi (p)-1}\parallel D{f}^m{|}_{E^s(f^{mi}(x))}\parallel \le C{\lambda }^{\pi (p)}\mathit{\text{and}}\hfill \\ \parallel D{f}^m{|}_{E^s(x)}\parallel \cdot \parallel D{f}^{-m}{|}_{E^u(f^m(x))}\parallel \le \lambda ,\hfill \end{array}$$

where $\pi (p)$ is the period of p.

Remark 2.12 By Pugh’s closing lemma, there is a residual set ${\mathcal{G}}_6\subset Diff(M)$ such that, for any $f\in {\mathcal{G}}_6$, if $f{|}_{\mathrm{\Lambda }}$ is transitive, then there is a periodic orbit $p_n$ such that $Orb(p_n)\to \mathrm{\Lambda }$ in Hausdorff metric.

Lemma 2.13 [[16], Theorem 3.8]

There is residual set ${\mathcal{G}}_5\subset Diff(M)$ such that, for any $f\in {\mathcal{G}}_5$, for any ergodic measure μ of f, there is a sequence of the periodic point $p_n$ such that ${\mu }_{p_n}\to \mu$ in weak^∗ topology and $Orb(p_n)\to Supp(\mu )$ in Hausdorff metric.

The following was proved by Mañé [17]. Denote by $\mathcal{M}(f{|}_{\mathrm{\Lambda }})$ the set of invariant probabilities on the Borel σ-algebra of Λ endowed with the weak^∗ topology.

Lemma 2.14 Let $\mathrm{\Lambda }\subset M$ be a closed f-invariant set of f and $E\subset T_{\mathrm{\Lambda }}M$ be a continuous invariant subbundle. If there is $m>0$ such that

$$\int log\parallel D{f}^m{|}_E\parallel d\mu <0$$

for every ergodic $\mu \in \mathcal{M}(f^m{|}_{\mathrm{\Lambda }})$, then E is contracting.

Proof of Theorem 1.4 Let $f\in {\mathcal{G}}_4\cap {\mathcal{G}}_5\cap {\mathcal{G}}_6$ have the ergodic shadowing property in a locally maximal Λ. Then by Proposition 2.11, we know that Λ admits a dominated splitting $T_{\mathrm{\Lambda }}M=E\oplus F$. Since f has the ergodic shadowing property in Λ, by Lemma 2.1, Remark 2.12, and Lemma 2.13, there is a sequence of periodic points such that $Orb(p_n)\to Supp(\mu )=\mathrm{\Lambda }$ in the Hausdorff metric. By Proposition 2.11, we have

$$\int \parallel D{f}^m{|}_E\parallel d\mu =\underset{n\to \mathrm{\infty }}{lim}\int \parallel D{f}^m{|}_E\parallel d{\mu }_{p_n}<0.$$

By Lemma 2.14, E is contracting. Similarly, we can show that F is expanding. □

Corollary 2.15 For $C^1$-generic f, if f has the ergodic shadowing property, then f is transitive Anosov.