A Type of the Shadowing Properties for Generic View Points

Abstract

We show that if a $C^1$ generic diffeomorphism of a closed smooth two-dimensional manifold has the average shadowing property or the asymptotic average shadowing property, then it is Anosov. Moreover, if a $C^1$ generic vector field of a closed smooth three-dimensional manifold has the average shadowing property or the asymptotic average shadowing property, then it satisfies singular Axiom A without cycles.

1. Introduction

The shadowing property is an important notion to study the stability systems in dynamical systems. Robinson [1] and Sakai [2] proved that a diffeomorphism f of a closed smooth manifold M has the $C^1$ robustly shadowing property if and only if it is structurally stable. However, Lewowicz [3] constructed examples of transitive diffeomorphisms with the shadowing property that are not Anosov. Thus, we know that the shadowing property does not imply hyperbolic systems. For these view points, Abdenur and Díaz [4] suggested the following problem; the shadowing property and hyperbolicity are equivalent in $C^1$ generic sense. Abdenur and Díaz [4] proved that, for a $C^1$ generic diffeomorphism $f:M\to M$, if a tame f has the shadowing property, then it is hyperbolic. However, if a $C^1$ generic diffeomorphism f is not tame, then the above problem is still open.

About this point, we consider a type of the shadowing property and hyperbolicity. The average shadowing property or the asymptotic average shadowing property are different types of the shadowing property. In fact, a Morse–Smale diffeomorphim has the shadowing property. However, the system has sinks or sources, and so it has neither the average shadowing property nor the asymptotic average shadowing property [5,6]. Many results were published about the relation between the shadowing properties and the hyperbolicity (see [1,2,7,8,9,10,11,12,13,14,15,16,17,18]). Among the results, we introduce some interesting results. Sakai [15] proved that if a diffeomorphism f of a surface has the $C^1$ robustly average shadowing property, then it is Anosov. Honary and Bahabadi [9] proved that if a diffeomorphism f of a surface has the $C^1$ robustly asymptotic average shadowing property, then it is Anosov. Lee and Park [10] proved that $C^1$ generically, if a diffeomorphism f has the average shadowing property or the asymptotic average shadowing property with every periodic points has intersection, then it is Anosov. Arbieto and Ribeiro [7] proved that if a vector field X of a closed smooth three-dimensional manifold has the $C^1$ robustly average shadowing property or the asymptotic average shadowing property, then it is Anosov. In any dimension, Ribeiro [16] proved that $C^1$ generically, if a vector field X has the average shadowing property or the asymptotic average shadowing property with every periodic orbits intersect each other, is Anosov. From these results, we show that if a diffeomorphism f or a vector field X has the average shadowing property or the asymptotic average shadowing property, then it is hyperbolic, in $C^1$ generic sense.

1.1. Diffeomorphisms

Let M be a compact two-dimensional manifold without boundary, and let $\mathrm{Diff}(M)$ be the space of $C^1$ diffeomorphisms of M. For any $\delta >0$, a sequence ${\{x_i\}}_{i\in \mathbb{Z}}\subset M$ is a δ-pseudo orbit of f if the distance between $f(x_i)$ and $x_{i+1}$ is less than $\delta$ for all $i\in \mathbb{Z}.$ For a closed f-invariant set $\mathsf{\Lambda }$, $f\in \mathrm{Diff}(M)$ has the shadowing property on $\mathsf{\Lambda }$ if, for every $ϵ>0,$ there is $\delta >0$ such that for any $\delta$-pseudo orbit ${\{x_i\}}_{i\in \mathbb{Z}}\subset \mathsf{\Lambda },$ we can choose a point $y\in M$ with the property $d(f^i(y),x_i)<ϵ$ for all $i\in \mathbb{Z}.$ If $\mathsf{\Lambda }=M$, then we say that f has the shadowing property.

Blank [19] introduced the notion of the average shadowing property. For $\delta >0$, a sequence ${\{x_i\}}_{i\in \mathbb{Z}}\subset M$ is a δ-average pseudo orbit of f if there is $K=K(\delta )>0$ such that, for all $n\ge K$ and $k\in \mathbb{Z},$ we have

$$\frac{1}{n}{\displaystyle \sum _{i=0}^{n-1}}d(f(x_{i+k}),x_{i+k+1})<\delta .$$

Definition 1.

Let $f\in \mathrm{Diff}(M).$ We say that f has the average shadowing property if for any $ϵ>0$ there is a $\delta >0$ such that every δ-average pseudo orbit ${\{x_i\}}_{i\in \mathbb{Z}}\subset M$ is ϵ-shadowed in average by the point $z\in M$, that is,

$$\underset{n\to \infty }{lim\; sup}\frac{1}{n}{\displaystyle \sum _{i=0}^{n-1}}d(f^i(z),x_i)<ϵand\underset{n\to -\infty }{lim\; sup}\frac{1}{n}{\displaystyle \sum _{i=0}^{n-1}}d(f^i(z),x_i)<ϵ.$$

Gu [5] introduced the asymptotic average shadowing property. A sequence ${\{x_i\}}_{i\in \mathbb{Z}}\subset M$ is an asymptotically average pseudo orbit of f if

$$\underset{n\to \infty }{lim}\frac{1}{2n+1}{\displaystyle \sum _{i=-n}^n}d(f(x_i),x_{i+1})=0.$$

An asymptotically pseudo orbit ${\{x_i\}}_{i\in \mathbb{Z}}\subset M$ is said to be asymptotically shadowed in average by the point z in M if

$$\underset{n\to \infty }{lim}\frac{1}{2n+1}{\displaystyle \sum _{i=-n}^n}d(f^i(z),x_i)=0.$$

Definition 2.

Let $f\in \mathrm{Diff}(M).$ We say that f has the asymptotic average shadowing property if every asymptotically average pseudo orbit $\{x_i\}\subset M$ of f can be asymptotically shadowed on average by some point in $M.$

Remark 1.

In [20], the authors proved that the asymptotic average shadowing property implies the average shadowing property for continuous surjective topological dynamical systems. For convenience, we will provide information on the results of the asymptotic average shadowing property.

A point $x\in M$ is chain recurrent if, for any $\eta >0$, there is a $\eta$-pseudo orbit ${\{x_i\}}_{i=0}^n(n\ge 1)$ such that $x_0=x$ and $x_n=x$. Let $\mathcal{CR}(f)$ be the set of all chain recurrent points of $f.$ An f-invariant closed set $C(f)\subset M$ is chain transitive if, for any $\delta >0$ and $x,y\in C(f)$, there is a $\delta$-pseudo orbit ${\{x_i\}}_{i=0}^n(n\ge 1)\subset C(f)$ such that $x_0=x$ and $x_n=y.$ We say that a diffeomorphism f is chain transitive if a chain transitive set $C(f)=M$.

Lemma 1.

Let $f\in \mathrm{Diff}(M)$. If f has one of the following,

(a) 

the average shadowing property (see [6]),

(b) 

the asymptotic average shadowing property (see [5]),

then f is chain transitive.

A point $p\in M$ is periodic if there is $n>0$ such that $f^n(p)=p$. Let $P(f)$ be the set of all periodic points of $f.$ We say that $p\in P(f)$ is a sink if all the eigenvalues of the derivative of p, that is, $D_p{f}^{\pi (p)},$ are less than one, and $p\in P(f)$ is a source if all the eigenvalues of $D_p{f}^{\pi (p)}$ are greater than one, where $\pi (p)$ is the period of $p.$

Lemma 2.

(Ref. [21] (Lemma 2.1)) If $C(f)$ is a chain transitive set of f, then $C(f)$ has neither sinks nor sources.

A subset $\mathcal{R}\subset \mathrm{Diff}(M)$ is said to be residual if it contains a countable intersection of open and dense subsets of $\mathrm{Diff}(M)$. A dynamic property is called $C^1$ generic if it holds in a residual subset of $\mathrm{Diff}(M).$ A closed f-invariant set $\mathsf{\Lambda }\subset M$ is transitive if there is $x\in \mathsf{\Lambda }$ such that $\omega (x)=\mathsf{\Lambda },$ where $\omega (x)$ is the omega limit set of $x.$ Crovisier [22] proved that for $C^1$ generic $f\in \mathrm{Diff}(M)$, a chain transitive set $C(X)$ is a transitive set $\mathsf{\Lambda }$. Then, according to Pugh’s closing lemma, there is a sequence of periodic orbit $P_n$ of f such that ${lim}_{n\to \infty }{P}_n\to \mathsf{\Lambda }$. If f is chain transitive, then we can consider the periodic point in $M.$ The result can be extended vector fields.

A closed f-invariant set $\mathsf{\Lambda }\subset M$ is hyperbolic if the tangent bundle $T_{\mathsf{\Lambda }}M$ has a $Df$-invariant splitting $E^s\oplus E^u$ and there exists $0<\lambda <1$ such that

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

for all $x\in \mathsf{\Lambda }$ and $n\ge 0.$ We say that f is Anosov if $\mathsf{\Lambda }=M$ in the above notion. We say that f is Axiom A if the nonwandering set $\mathsf{\Omega }(f)$ is the closure of $P(f)$ and is hyperbolic. According to Mañé [23], we have the following.

Theorem 1.

(Ref. [24] (Theorem(Mañé)) There is a residual subset $\mathcal{R}\subset \mathrm{Diff}(M)$ such that for $f\in \mathcal{R}$ satisfies one of the following:

(a) 

f is Axiom A without cycles;

(b) 

f has infinitely many sinks or sources.

From Theorem 1, we have the following theorem, which is a main result of the paper.

Theorem 2.

There is a residual subset $\mathcal{R}\subset \mathrm{Diff}(M)$ such that for $f\in \mathcal{R}$, if f has the following;

(a) 

the asymptotic average shadowing property, or

(b) 

the average shadowing property,

then f is Anosov.

Proof. 

Let $f\in \mathcal{R}$ have the asymptotic average shadowing property or the average shadowing property. According to Lemma 1, f is chain transitive. By Lemma 2, f has neither sinks nor sources. According to Theorem 1, f satisfies Axiom A without cycles. Finally, we show that $\mathsf{\Omega }(f)=M.$ Since f is Axiom A without cycles, we have the nonwandering set $\mathsf{\Omega }(f)$ is hyperbolic and $\mathsf{\Omega }(f)=\mathcal{CR}(f).$ Since f is chain transitive, $\mathcal{CR}(f)=M$ and $\mathcal{CR}(f)$ is hyperbolic, and so f is Anosov. ☐

In the introduction, we mention that the asymptotic average shadowing property or the average shadowing property is not equal to the shadowing property. However, $C^1$ generically, in three-dimensional cases, if a diffeomorphism f has the asymptotic average shadowing property or the average shadowing property, then it has the shadowing property.

Corollary 1.

For $C^1$ generic $f\in \mathrm{Diff}(M)$, if f has the asymptotic average shadowing property or the average shadowing property, then it has the shadowing property.

Proof. 

Let $f\in \mathcal{R}$ have the average shadowing property or the asymptotic average shadowing property. According to Theorem 2, f is Anosov. By [25], f has the shadowing property. ☐

1.2. Vector Fields

Let M be a compact three-dimensional manifold without boundaries. Denote by $\mathfrak{X}(M)$ the set of $C^1$-vector fields on M endowed with the $C^1$-topology. Let $X\in \mathfrak{X}(M)$. The flow of X will be denoted by $X^t,t\in \mathbb{R}.$ A point $x\in M$ is singular of X if $X^t(x)=x$ for all $t\in \mathbb{R}.$ Denote by $Sing(X)$ the set of all singular points of X. A point $p\in M$ is periodic if there is $\pi (p)>0$ such that $X^{\pi (p)}(p)=p$, where $\pi (p)$ is the prime period of $p.$ Let $Per(X)$ be the set of all closed orbits of X. Let $Crit(X)=Sing(X)\cup Per(X).$ It is clear that $Crit(X)\subset \mathsf{\Omega }(X)$, where $\mathsf{\Omega }(X)$ is the set of all nonwandering points of X. For any $\delta >0$, a sequence $\{(x_i,t_i):x_i\in M,t_i\ge 1\mathrm{and}i\in \mathbb{Z}\}$ is a $\delta$-pseudo orbit of X if $d(X^{t_i}(x_i),x_{i+1})<\delta$ for any $i\in \mathbb{Z}.$ For vector fields, the average shadowing property was introduced by Gu et al. [26] and the asymptotic average shadowing property was introduced by Gu [27]. We say that a homeomorphism $h:\mathbb{R}\to \mathbb{R}$ is a reparametrization of $\mathbb{R}$ if h is increasing and $h(0)=0.$ Denote by Rep($\mathbb{R}$) the set of reparametrizations of $\mathbb{R}.$ For $ϵ>0$, we define Rep$(ϵ)$ as follows:

$$\mathrm{Rep}(ϵ)=\left\{h\in \mathrm{Rep}(\mathbb{R}):|\frac{h(t)}{t}-1|<ϵ\right\}.$$

For any $\delta >0$, a sequence $\{(x_i,t_i):x_i\in M,t_i\ge 1,\mathrm{and}i\in \mathbb{Z}\}$ is a δ-average pseudo orbit of X if there is a $T>0$ such that for any $n\ge T$ and $k\in \mathbb{Z}$, we have

$$\frac{1}{n}{\displaystyle \sum _{i=0}^{n-1}}d(X^{t_{i+k}}(x_{i+k}),x_{i+k+1})dt<\delta .$$

A $\delta$-average pseudo orbit ${\{(x_i,t_i)\}}_{i\in \mathbb{Z}}\subset M$ is positively shadowed on average by the orbit of X through a point $z\in M$ if there exists $h\in \mathrm{Rep}(ϵ)$ with $h(0)=0$ such that

$$\underset{n\to \infty }{lim\; sup}\frac{1}{n}{\displaystyle \sum _{i=0}^{n-1}}{\int }_{s_i}^{s_{i+1}}d(X^{h(t)}(z),X^{t-s_i}(x_i))dt<ϵ,$$

where $s_0=0$, and $s_n={\sum }_{i=0}^{n-1}{t}_i(i\in \mathbb{N}).$ Analogously, we define negatively shadowed on average.

Definition 3.

Let $X\in \mathfrak{X}(M)$. We say that X has the average shadowing property if, for every $ϵ>0$, there is a $\delta >0$ such that every δ-average pseudo orbit ${\{(x_i,t_i)\}}_{i\in \mathbb{Z}}\subset M$ of X can be positively and negatively shadowed in average by the orbit of X through some point in M.

A sequence $\{(x_i,t_i):x_i\in M,t_i\ge 1\mathrm{and}i\in \mathbb{Z}\}$ is an asymptotically average pseudo orbit of X if

$$\underset{n\to \infty }{lim}\frac{1}{2n+1}{\displaystyle \sum _{i=-n}^n}d(X^{t_i}(x_i),x_{i+1})=0.$$

An asymptotically average pseudo orbit ${\{(x_i,t_i)\}}_{i\in \mathbb{Z}}$ is positively asymptotically shadowed on average by the orbit of X through a point $z\in M$ if there is $h\in \mathrm{Rep}(ϵ)$ with $h(0)=0$ such that

$$\underset{n\to \infty }{lim}\frac{1}{n}{\displaystyle \sum _{i=0}^{n-1}}{\int }_{s_i}^{s_{i+1}}d(X^{h(t)}(z),X^{t-s_i}(x_i))dt=0,$$

where $s_0=0$, and $s_n={\sum }_{i=0}^{n-1}{t}_i(i\in \mathbb{N}).$ Analogously, we define negatively asymptotically shadowed on average.

Definition 4.

Let $X\in \mathfrak{X}(M)$. We say that X has the asymptotic average shadowing property if every asymptotically average pseudo orbit ${\{(x_i,t_i)\}}_{i\in \mathbb{Z}}\subset M$ of X can be positively and negatively asymptotically shadowed on average by some point in M.

A closed $X^t$-invariant set $\mathsf{\Lambda }$ is transitive if $\overline{Orb(x)}=\mathsf{\Lambda }$ for some $x\in \mathsf{\Lambda }$, where $\overline{A}$ is the closure of set $A.$ A compact $X^t$-invariant $C(X)\subset M$ is chain transitive if, for any $x,y\in C(X)$ and $\eta >0$, there is a $\eta$-pseudo orbit $\{(x_i,t_i):t_i\ge 1,for0\le i\le n\}(n\ge 1)\subset C(X)$ with $x_0=x,x_n=y$. Note that if a flow is transitive, then it is chain transitive. However, the converse is not true in general. A point $x\in M$ is chain recurrent if, for any $\eta >0,$ there is a finite $\eta$-pseudo orbit from x to x. Let $\mathcal{CR}(X)$ be the set of all chain recurrent points of $X.$

Lemma 3.

Let $X\in \mathfrak{X}(M)$. If X has one of the following,

(a) 

the average shadowing property (see [26]),

(b) 

the asymptotic average shadowing property (see [27]),

then X is chain transitive.

A periodic point p is a sink if the eigenvalues of $D_{p}f$ have absolute values of less than one, where $D_{p}f$ is the derivative of p and f is the Poincaré map of $X.$ A source is sink for a vector field $-X.$

Lemma 4.

(Ref. [7] (Lemma 6)) If $C(X)$ is a chain transitive set of X, then $C(X)$ has neither sinks nor sources.

A compact $X^t$ invariant set $\mathsf{\Lambda }$ is called hyperbolic for $X^t$ if there are constants $C>0,\lambda >0$ and a splitting $T_{x}M=E_x^s\oplus ⟨X(x)⟩\oplus E_x^u$ such that the tangent flow $D{X}^t:TM\to TM$ leaves the invariant continuous splitting and

$$\parallel D{X}^t{|}_{E_x^s}\parallel \le C{e}^{-\lambda t}\mathrm{and}\parallel D{X}^{-t}{|}_{E_x^u}\parallel \le C{e}^{-\lambda t}$$

for $t>0$ and $x\in \mathsf{\Lambda },$ where $⟨X(x)⟩$ is the subspace generated by X. We say that $X\in \mathfrak{X}(M)$ is Anosov if $M=\mathsf{\Lambda }$ in the above notion.

We say that $\mathsf{\Lambda }$ is partially hyperbolic if there is an invariant splitting $T_{\mathsf{\Lambda }}M=E^s\oplus E^c$ and constants $C>0,\lambda >0$ such that

(i)

$\parallel D{X}^t{|}_{E_x^s}\parallel \le C{e}^{-\lambda t}$, for all $x\in \mathsf{\Lambda }$ and $t>0,$

(ii)

$E^s$ dominates $E^c$, that is, $E_x^s\ne 0,E_x^c\ne 0$ and $\parallel D{X}^t{|}_{E_x^s}\parallel ·\parallel D{X}^{-t}{|}_{E_{X_t(x)}^c}\parallel \le C{e}^{-\lambda t}$, for all $x\in \mathsf{\Lambda }$ and $t>0.$

In the above definition, we say that the central bundle $E^c$ is volume expanding if the constants $C>0$ and $\lambda >0$ satisfy $|J(D{X}^t{|}_{E_x^c})|\ge C{e}^{\lambda t},$ for all $x\in \mathsf{\Lambda }$ and $t>0$, where $J(·)$ is the Jacobian.

A compact $X^t$-invariant $\mathsf{\Lambda }$ is attracting if there is a neighborhood U of $\mathsf{\Lambda }$ such that $\mathsf{\Lambda }={\bigcap }_{t\ge 0}{X}^t(U).$ A set $\mathsf{\Lambda }$ is an attractor if it is a transitive attracting set.

We say that $\mathsf{\Lambda }$ is singular hyperbolic if every singularity in $\mathsf{\Lambda }$ is hyperbolic and it is partially hyperbolic with a volume expanding central bundle. A singular hyperbolic attractor is an attractor that is also a singular hyperbolic set for X, and a singular hyperbolic repeller is an attractor that is a singular hyperbolic set for $-X.$ We say that X satisfies Axiom A if the nonwandering set $\mathsf{\Omega }(X)=\overline{Crit(X)}$ is hyperbolic. Note that if X satisfies Axiom A, then we know that $\mathsf{\Omega }(X)={\mathsf{\Lambda }}_1\cup \cdots \cup {\mathsf{\Lambda }}_n$, where each ${\mathsf{\Lambda }}_i$ is a hyperbolic basic set of X (see [28])). A collection of basic sets ${\mathsf{\Lambda }}_{i_1},\dots ,{\mathsf{\Lambda }}_{i_k}$ of X is called a cycle, if there exist points $x_n\in \mathsf{\Omega }(X)(1\le n\le k$) such that $\alpha (x_n)\in {\mathsf{\Lambda }}_{i_n}$ and $\omega (x_n)\in {\mathsf{\Lambda }}_{i_{n+1}}(k+1\equiv 1).$ An Axiom A vector field X satisfies the no-cycle condition if there are no cycles among the basic sets of X. A vector field X is called singular Axiom A if there is a finite disjoint decomposition $\mathsf{\Omega }(X)={\mathsf{\Lambda }}_1\cup \cdots \cup {\mathsf{\Lambda }}_n,$ where each ${\mathsf{\Lambda }}_i$ is a hyperbolic basic set, a singular hyperbolic attractor or a singular hyperbolic repeller, $i=1,\dots ,n.$

A subset $\mathcal{G}\subset \mathfrak{X}(M)$ is called residual if it contains a countable intersection of open and dense subsets of $\mathfrak{X}(M)$. A dynamic property is called $C^1$ generic if it holds in a residual subset of $\mathfrak{X}(M).$ The following is found in ([29] (Thereom A)).

Theorem 3.

There is a residual subset $\mathcal{G}\subset \mathfrak{X}(M)$ such that for $X\in \mathcal{G}$, X satisfies one of the following:

(a) 

X is singular Axiom A without cycles;

(b) 

X has infinitely many sinks or sources.

According to Theorem 3, we have the following result, which is a main result in the paper and is a vector field version of Theorem 2.

Theorem 4.

There is a residual subset $\mathcal{G}\subset \mathfrak{X}(M)$ such that, for $X\in \mathcal{G}$, if X has the following:

(a) 

asymptotic average shadowing property, or

(b) 

average shadowing property,

then X is singular Axiom A without cycles.

Proof. 

Let $X\in \mathcal{G}$ have the average shadowing property or the asymptotic average shadowing property. According to Lemma 3, X is chain transitive. Since X is chain transitive, according to Lemma 4, X has neither sinks nor sources. Since $X\in \mathcal{G},$ by Theorem 3, X is singular Axiom A without cycles. ☐

Corollary 2.

There is a residual subset $\mathcal{G}\subset \mathfrak{X}(M)$ such that, for $X\in \mathcal{G}$, if $Sing(X)=\varnothing$ and X has the average shadowing property or the asymptotic average shadowing property, then X is Anosov.

Proof. 

Let $X\in \mathcal{G}$ have the average shadowing property or the asymptotic average shadowing property. As in the proof of Theorem 4, X is singular Axiom A without cycles. Since $Sing(X)=\varnothing$, X is Axiom A without cycles. Since X is chain transitive, the chain recurrence set $\mathcal{CR}(X)=M$. Since X is Axiom A without cycles and $Sing(X)=\varnothing$, by ([30] (Theorem A)), $\mathcal{CR}(X)=\mathsf{\Omega }(X)$. Thus, X is Anosov.  ☐

The following is similar to the proof Corollary 1. Thus, we omit the proof.

Corollary 3.

There is a residual subset $\mathcal{G}\subset \mathfrak{X}(M)$ such that, for $X\in \mathcal{G}$, if $Sing(X)=\varnothing$ and X has the average shadowing property or the asymptotic average shadowing property, then X has the shadowing property.

Remark 2.

In Corollary 3, for $C^1$ generic $X\in \mathfrak{X}(M)$, if $Sing(X)\ne \varnothing$, then the result is still open (see [17]). Thus, we will consider that, $C^1$ generically, if $Sing(X)\ne \varnothing$, then does X have the shadowing property?