Filtrated Pseudo-Orbit Shadowing Property and Approximately Shadowable Measures

Abstract

In this paper, it is proved that every diffeomorphism possessing the filtrated pseudo-orbit shadowing property admits an approximately shadowable Lebesgue measure. Furthermore, the $C^1$-interior of the set of diffeomorphisms possessing the filtrated pseudo-orbit shadowing property is characterized as the set of diffeomorphisms satisfying both Axiom A and the no-cycle condition. As a corollary, it is proved that there exists a $C^1$-open set of diffeomorphisms, any element of which does not have the shadowing property but admits an approximately shadowable Lebesgue measure.

1. Introduction

The notion of pseudo-orbits appears often in the several branches of the modern theory of dynamical systems; especially, the pseudo-orbit shadowing property usually plays an important role in the investigation of stability theory and ergodic theory. Let $(X,d)$ be a compact metric space, and let $f:X\to X$ be a homeomorphism. For $\delta >0$, a sequence of points ${\left\{x_i\right\}}_{i=a}^b\subset X$$(-\infty \le a<b\le \infty )$ is called a δ-pseudo-orbit of f if $d(f\left(x_i\right),x_{i+1})<\delta$ for all $a\le i\le b-1$.

Denote by $f_{|A}$ the restriction of f to a set $A\subset X$. Let $\Lambda \subset X$ be a closed set (not necessarily f-invariant). We say that $f_{|\Lambda }$ has the shadowing property if for every $ϵ>0$ there is $\delta >0$ such that for any $n\in \mathbb{N}$ and $\delta$-pseudo-orbit ${\left\{x_i\right\}}_{i=0}^{n-1}\subset \Lambda$ of f there is $y\in X$$ϵ$-shadowing the pseudo-orbit—that is, $d(f^i\left(y\right),x_i)<ϵ$ for all $0\le i\le n-1$. Note that only $\delta$-pseudo-orbits of f “contained in $\Lambda$” can be $ϵ$-shadowed, but the shadowing point $y\in X$ is “not necessarily” contained in $\Lambda$. We say that f has the shadowing property if $X=\Lambda$ in the above definition. Since X is compact, it is not difficult to show that if $f_{|\Lambda }$ has the shadowing property, then every pseudo-orbit ${\left\{x_i\right\}}_{i=-\infty }^{\infty }\subset \Lambda$ can be shadowed by some true orbit of f.

In [1], we introduced the notion of shadowable measures as a generalization of the shadowing property from the measure theoretical view point, and investigated the dynamics of diffeomorphisms satisfying the notion (in fact, the dynamics of the $C^1$-interior of the set of diffeomorphisms possessing the shadowable measures is characterized as uniform hyperbolicity—see [1], Theorems 1 and 2). Every dynamical system possessing the shadowing property admits shadowable measures, but the converse is not generally true. In fact, an example of a diffeomorphism g is constructed on the 2-torus ${\mathbb{T}}^2$ such that g does not have the shadowing property but admits a shadowable Lebesgue measure (see [1], Example 3).

In this paper, generalizing the dynamics and shadowable Lebesgue measure of this example on ${\mathbb{T}}^2$, we introduce the notion of the filtrated pseudo-orbit shadowing property and that of approximately shadowable measures, and we prove that every diffeomorphism having the property admits an approximately shadowable Lebesgue measure. Furthermore, the $C^1$-interior of the set of diffeomorphisms possessing the filtrated pseudo-orbit shadowing property is characterized as the set of diffeomorphisms satisfying both Axiom A and the no-cycle condition. Finally, by making use of a quasi-Anosov diffeomorphism, we construct a $C^1$-open set of diffeomorphisms, any element of which does not have the shadowing property but admits an approximately shadowable Lebesgue measure.

2. Definitions and Statement of the Results

Recall that $(X,d)$ is a compact metric space and $f:X\to X$ is a homeomorphism of X. For given points $x,y\in X$ and $\delta >0$, we write $x{⇝}^{\delta }y$ if there is a $\delta$-pseudo-orbit ${\left\{x_i\right\}}_{i=0}^n$ of f such that $x_0=x$ and $x_n=y$ for some $n=n_{\delta }\in \mathbb{N}$. Write $x{\leftrightsquigarrow }^{\delta }y$ if $x{⇝}^{\delta }y$ and $y{⇝}^{\delta }x$. Finally, we write $x\leftrightsquigarrow y$ if $x{\leftrightsquigarrow }^{\delta }y$ for any $\delta >0$. The chain recurrent set of f, denoted by $\mathcal{R}\left(f\right)$, is the set of points $x\in X$ such that $x\leftrightsquigarrow x$. The chain recurrent set is one of the main subjects to consider in the shadowing theory of dynamical systems. Clearly, $\Omega \left(f\right)\subset \mathcal{R}\left(f\right)$ by definition, where $\Omega \left(f\right)$ is the non-wandering set of f.

Let $X^n=X\times \cdots \times X$ (the n-times of direct product) be the sequences of points of X with length $n\in \mathbb{N}$, and denote by $\mathcal{M}\left(X\right)$ the space of Borel probability measures of X. For any $\mu \in \mathcal{M}\left(X\right)$ (not necessarily f-invariant), let ${\mu }_n=\mu \times \cdots \times \mu$ (n-times) be the direct product measure of $X^n$. For any $\delta >0$, denote by $\mathcal{PO}(\delta ,n)$ the space of $\delta$-pseudo-orbits ${\left\{x_i\right\}}_{i=0}^{n-1}\in X^n$ of f, and for $ϵ>0$, denote by $\mathcal{SPO}(\delta ,ϵ,n)(\subset \mathcal{PO}(\delta ,n\left)\right)$ the set of $\delta$-pseudo-orbits $ϵ$-shadowed by some point.

We say that $\mu \in \mathcal{M}\left(X\right)$ is a shadowable measure of f (or simply, f is $\mu$-shadowable) if for any $ϵ>0$ there exists $\delta >0$ such that

$${\mu }_n\left(\mathcal{SPO}(\delta ,ϵ,n)\right)={\mu }_n\left(\mathcal{PO}(\delta ,n)\right)$$

for any $n\in \mathbb{N}$ (if A is a subset of X, then we define the shadowable measure for $f_{|A}$ by the same manner). Observe that if f has the shadowing property, then f is $\mu$-shadowable for any $\mu \in \mathcal{M}\left(X\right)$. Denote by $\mathrm{supp}\left(\mu \right)$ the support of $\mu \in \mathcal{M}\left(X\right)$. Then, since X is compact, it is known that if f is $\mu$-shadowable, then $f_{\left|\mathrm{supp}\right(\mu )}$ has the shadowing property (see [1], Lemma 1).

In this paper, we generalize the notion of shadowable measures to describe the dynamics of the system such as ([1], Example 3) from the measure theoretical view point. Let $\mu \in \mathcal{M}\left(X\right)$, and let $Y\subset X$ be a Borel set. For any Borel set $A\subset X$, we put

$$\tilde{\mu }\left(A\right)={\tilde{\mu }}_Y\left(A\right)=\frac{\mu (A\cap Y)}{\mu \left(Y\right)}\in \mathcal{M}\left(X\right).$$

We say that $\mu$ is approximately shadowable if for any $ϵ>0$ there exists a Borel set Y of X with $\mu \left(Y\right)>1-ϵ$ and $f\left(Y\right)\subset Y$ such that $\tilde{\mu }$ is a shadowable measure

Hereafter, let M be a closed $C^{\infty }$ manifold, and let d be a distance on M induced from a Riemannian metric $\parallel ·\parallel$ on the tangent bundle $TM$. Denote by $\mathrm{Diff}\left(M\right)$ the space of diffeomorphisms of M endowed with the $C^1$-topology as usual. We say that a sequence

$$\varnothing =M_0\subset M_1\subset \cdots \subset M_K=M$$

of smooth compact submanifolds $M_k$ with boundary such that $dim{M}_k=dimM$ for $1\le k\le K$ is a filtration adapted to$f\in \mathrm{Diff}\left(M\right)$ if the following conditions $\left(a\right)$ and $\left(b\right)$ are met:

(a) 

The chain recurrent set $\mathcal{R}\left(f\right)$ of f is composed of mutually disjoint closed f-invariant sets ${\left\{{\Lambda }_k\left(f\right)\right\}}_{k=1}^K$$(K\ge 1)$ of f—that is,

$$\mathcal{R}\left(f\right)={\Lambda }_1\left(f\right)\cup \cdots \cup {\Lambda }_K\left(f\right);$$

(b) 

For any $1\le k\le K$,

(b.1) 

$f\left(M_k\right)\subset \mathrm{int}{M}_k$;

(b.2) 

${\Lambda }_k\left(f\right)\subset \mathrm{int}(M_k\setminus M_{k-1})$;

(b.3) 

${\Lambda }_k\left(f\right)={\bigcap }_{m\in \mathbb{Z}}{f}^m(M_k\setminus M_{k-1})$.

Here $\mathrm{int}A$ denotes the interior of a set $A\subset M$.

We say that f has the filtrated pseudo-orbit shadowing property if there exists a filtration $\varnothing =M_0\subset M_1\subset \cdots \subset M_K=M$$(K\ge 1)$ adapted to f such that for all $ϵ>0$ there exists $\delta >0$ such that for any filtrated $\delta$-pseudo-orbit ${\left\{x_i\right\}}_{i=0}^n\subset \mathrm{int}(M_k\setminus M_{k-1})$ of f$(n\in \mathbb{N})$ there exists $x\in M$ satisfying $d(f^i\left(x\right),x_i)<ϵ$ for all $0\le i\le n$.

Denote by $\mathcal{FS}$ the set of $f\in \mathrm{Diff}\left(M\right)$ having the filtrated pseudo-orbit shadowing property. The first result of this paper is the following.

Theorem 1.

Every $f\in \mathcal{FS}$ admits an approximately shadowable Lebesgue measure $m\in \mathcal{M}\left(M\right)$.

Remark 1.

Suppose $m\in \mathcal{M}\left(M\right)$ is the normalized Lebesgue measure on M. Let us emphasize at this point that this m is an approximately shadowable Lebesgue measure for $f\in \mathcal{FS}$. In fact, we will see that for any $ϵ>0$ there exists a set $Y\subset M$$\left(m\right(Y)>1-ϵ)$ such that $\tilde{m}\in \mathcal{M}\left(M\right)$ is shadowable and $\mathrm{supp}\left(\tilde{m}\right)=Y$.

Denote by $\mathrm{int}\mathcal{FS}$ the $C^1$-interior of the set $\mathcal{FS}$ in $\mathrm{Diff}\left(M\right)$; that is, $f\in \mathrm{int}\mathcal{FS}$ if and only if there exists a $C^1$-neighborhood $\mathcal{U}\left(f\right)$ of f in $\mathrm{Diff}\left(M\right)$ such that any $g\in \mathcal{U}\left(f\right)$ meets all the properties $\left(a\right)$, $\left(b\right)$ with respect to g and has the filtrated pseudo-orbit shadowing property. More precisely, for any $g\in \mathcal{U}\left(f\right)$,

-

$\mathcal{R}\left(g\right)$ is composed of mutually disjoint closed g-invariant sets ${\left\{{\Lambda }_k\left(g\right)\right\}}_{k=1}^{K_g}$$(K_g\ge 1)$—that is,

$$\mathcal{R}\left(g\right)={\Lambda }_1\left(g\right)\cup \cdots \cup {\Lambda }_{K_g}\left(g\right)$$

and properties $(b.1)$–$(b.3)$ are met, and

-

Any filtrated pseudo-orbit ${\left\{x_i\right\}}_{i=0}^n\subset \mathrm{int}(M_k^g\setminus M_{k-1}^g)$ of g$(n\in \mathbb{N})$ is g-shadowed—that is, $g_{\left|\mathrm{int}\right(M_k^g\setminus M_{k-1}^g)}$ has the shadowing property for $1\le k\le K_g$.

Let $\Lambda \subset M$ be a closed f-invariant set. The set $\Lambda$ is hyperbolic if the tangent bundle $T_{\Lambda }M$ has a $Df$-invariant splitting $E^s\oplus E^u$ with constants $C>0$ and $0<\lambda <1$ such that

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

for all $x\in \Lambda$ and $n\ge 0$. Suppose that $\Lambda$ is hyperbolic. Then it is well-known that $f_{|\Lambda }$ has the shadowing property (see [2,3]).

The stable manifold of a point $x\in \Lambda$ is the set

$$W^s\left(x\right)=\{y\in M:d(f^n\left(x\right),f^n\left(y\right))\to 0\mathrm{as}n\to \infty \}.$$

The unstable manifold, $W^u\left(x\right)$, of $x\in \Lambda$ is also defined analogously for $n\to -\infty$. It is also well-known that $W^s\left(x\right)$ and $W^u\left(x\right)$ are both immersed manifolds (see [3], among others).

We say that f is Anosov when the whole space M is hyperbolic. At this moment, let us remark that any $\mu \in \mathcal{M}\left(M\right)$ is shadowable if f is Anosov, and thus, every Anosov diffeomorphism admits a shadowable Lebesgue measure m such that

$$\mathrm{supp}\left(m\right)=M.$$

Hereafter, let $P\left(f\right)$ be the set of periodic points of f, and recall that $\Omega \left(f\right)$ is the set of non-wandering points of it. We say that f satisfy Axiom A if $\Omega \left(f\right)$ is hyperbolic and $\Omega \left(f\right)=\overline{P\left(f\right)}$. Let f satisfies Axiom A. Then the non-wandering set has the so-called spectral decomposition—that is,

$$\Omega \left(f\right)={\Lambda }_1\left(f\right)\cup \cdots \cup {\Lambda }_L\left(f\right)$$

composed of basic sets ${\left\{{\Lambda }_i\left(f\right)\right\}}_{i=1}^L$, and satisfies

$$M=\bigcup _{i=1}^L{W}^{\sigma }\left({\Lambda }_i\left(f\right)\right),$$

where

$$W^{\sigma }\left({\Lambda }_i\left(f\right)\right)={\cup }_{x\in {\Lambda }_i\left(f\right)}{W}^{\sigma }\left(x\right)(\sigma =s,u)$$

for $1\le i\le L$.

We say that f has a cycle if there is a subsequence ${\left\{{\Lambda }_{i_j}\left(f\right)\right\}}_{j=1}^l$$(2\le l\le L+1)$ of the spectral decomposition such that

$${\Lambda }_{i_1}\left(f\right)={\Lambda }_{i_l}\left(f\right)\mathrm{and}{W}^u\left({\Lambda }_{i_j}\left(f\right)\right)\cap W^s\left({\Lambda }_{i_{j+1}}\left(f\right)\right)\ne \varnothing (1\le j\le l-1).$$

Note that if f satisfies the no-cycle condition, then $\mathcal{R}\left(f\right)=\Omega \left(f\right)$ (see [4]).

The next result is the following.

Theorem 2.

Let $f\in \mathrm{Diff}\left(M\right)$. Then $f\in \mathrm{int}\mathcal{FS}$ if and only if f satisfies both Axiom A and the no-cycle condition.

We say that f is quasi-Anosov if for any $v\in TM\setminus \left\{0\right\}$, $\{\parallel D{f}^n\left(v\right)\parallel :n\in \mathbb{Z}\}$ is unbounded. In [5], quasi-Anosov diffeomorphisms are characterized as diffeomorphisms satisfying both Axiom A and the no-cycle condition such that for any $x\in M$,

$$T_x{W}^s\left(x\right)\cap T_x{W}^u\left(x\right)=\left\{0\right\}.$$

Every Anosov diffeomorphism is quasi-Anosov, but an example of quasi-Anosov, non-Anosov diffeomorphism is constructed by [6] (for more information, see [7]). In this paper, by making use of a quasi-Anosov diffeomorphism, we construct a $C^1$-open set of $\mathrm{Diff}\left(M\right)$, any element of which does not have the shadowing property but admits an approximately shadowable Lebesgue measure (see Corollary 1).

For quasi-Anosov diffeomorphisms, the relationship to the shadowing property is considered in [8], and the following result is obtained therein.

Theorem 3.

Let $f\in \mathrm{Diff}\left(M\right)$. Then f is quasi-Anosov possessing the shadowing property if and only if f is Anosov.

Since every quasi-Anosov diffeomorphism is in $\mathrm{int}\mathcal{FS}$ by Theorem 2, the next result follows from Theorems 1 and 3.

Corollary 1.

Let $f\in \mathrm{Diff}\left(M\right)$ be a quasi-Anosov diffeomorphism that is not Anosov. Then there is a $C^1$-open set (a neighborhood of f), any g of which does not have the shadowing property but admits an approximately shadowable Lebesgue measure m.

Remark 2.

Since the example g on ${\mathbb{T}}^2$ constructed in ([1], Example 3) is in $\mathrm{int}\mathcal{FS}$, every h$C^1$-nearby g admits an approximately shadowable Lebesgue measure. However, it is easy to see that, for any $C^1$-neighborhood $\mathcal{V}\left(g\right)$ of g, there is a $h\in \mathcal{V}\left(g\right)$ possessing the shadowing property.

We close this section by pointing out an example which does not admit an approximately shadowable Lebesgue measure.

Example 1.

Let $S^1=\{e^{2\pi i\theta }:\theta \in \mathbb{R}\}\subset \mathbb{C}$. For $\alpha \in \mathbb{R}\setminus \mathbb{Q}$, let ${\rho }_{\alpha }:S^1\to S^1$ be an irrational rotation map defined by ${\rho }_{\alpha }\left(e^{2\pi i\theta }\right)=e^{2\pi i(\theta +\alpha )}$. Then the map ${\rho }_{\alpha }$ does not admit an approximately shadowable Lebesgue measure since ${\rho }_{\alpha }$ does not satisfy the shadowing property (see [1], Example 2) and we have $\overline{Y}=S^1$ for $\varnothing \ne Y\subset S^1$ with $f\left(Y\right)\subset Y$.

3. Proofs of the Results

In this section, we give the proofs of Theorems 1 and 2 and Corollary 1.

Proof of Theorem 1. 

Suppose that $f\in \mathcal{FS}$, and let $m\in \mathcal{M}\left(M\right)$ be the normalized Lebesgue measure on M. Let $\varnothing =M_0\subset M_1\subset \cdots \subset M_K=M$ be a filtration adapted to f as in the definition of the filtrated pseudo-orbit shadowing property, and recall the conditions $\left(a\right)$ and $\left(b\right)$ that f meets. We define a stable set for ${\Lambda }_k\left(f\right)$$(1\le k\le K)$ by

$$W^s\left({\Lambda }_k\left(f\right)\right)=\{y\in M:d(f^n\left(y\right),{\Lambda }_k\left(f\right))\to 0\mathrm{as}n\to \infty \}.$$

Clearly, we have $f\left(W^s\left({\Lambda }_k\left(f\right)\right)\right)=W^s\left({\Lambda }_k\left(f\right)\right)$ and $W^s\left({\Lambda }_k\left(f\right)\right)\cap W^s\left({\Lambda }_l\left(f\right)\right)=\varnothing$ for $k\ne l$.

By $(b.1)$–$(b.3)$ we have

(1)

$W^s\left({\Lambda }_k\left(f\right)\right)\cap M_{k-1}=\varnothing$;

(2)

${\bigcap }_{n=0}^{\infty }{f}^{-n}(M_k\setminus M_{k-1})=W^s\left({\Lambda }_k\left(f\right)\right)\cap M_k$;

(3)

${\bigcup }_{k=1}^K{W}^s\left({\Lambda }_k\left(f\right)\right)=M$.

If we set

$$W_k^n=W^s\left({\Lambda }_k\left(f\right)\right)\cap f^{-n}\left(M_k\right)$$

for $1\le k\le K$ and $n\ge 0$, then ${\left\{W_k^n\right\}}_{n=0}^{\infty }$ is an increasing sequence of closed sets satisfying that for $1\le k\le K$

(4)

$f\left(W_k^n\right)=W_k^{n-1}\subset W_k^n$, $f^n\left(W_k^n\right)=W_k^0\subset M_k\setminus M_{k-1}$;

(5)

$W_k^n\cap W_l^n=\varnothing$$(k\ne l)$;

(6)

${\Lambda }_k\left(f\right)\subset W_k^0\subset W_k^1\subset \cdots \subset W^s\left({\Lambda }_k\left(f\right)\right)={\bigcup }_{n=0}^{\infty }{W}_k^n.$

Then, by (3) and (6), for $ϵ>0$ there exists $N\in \mathbb{N}$ such that

$$m\left(\bigcup _{k=1}^K{W}_k^N\right)>1-ϵ.$$

In what follows, we put

$$Y=\bigcup _{k=1}^K{W}_k^N$$

for convenience (note that $f\left(Y\right)\subset Y$ by (4)).

Now, let us take ${\delta }_1>0$ such that if $x\in W_k^N$ and $y\in W_l^N$ for $k\ne l$, then $d(x,y)>{\delta }_1$ (see (5) above). Thus we have the following

Claim. There exists $0<{\delta }_2<{\delta }_1$ such that for any $\delta$-pseudo-orbit ${\left\{x_i\right\}}_{i=0}^n\subset Y$ of f with $0<\delta <{\delta }_2$ and $n\ge N+1$, there is $1\le k\le K$ such that $x_i\in \mathrm{int}(M_k\setminus M_{k-1})$ for all $N+1\le i\le n$.

Indeed, let ${\left\{x_i\right\}}_{i=0}^n\subset Y$ be a given $\delta$-pseudo-orbit of f with $0<\delta <{\delta }_1$ and $n\ge N+1$. Then it is easy to see that by the choice of ${\delta }_1>0$, there is $1\le k\le K$ such that ${\left\{x_i\right\}}_{i=0}^n\subset W_k^N$. Since

$$f^{N+1}\left(W_k^N\right)\subset f\left(M_k\right)\subset \mathrm{int}{M}_k,$$

by the uniform continuity of f, we can choose $0<{\delta }_2<{\delta }_1$ such that if $0<\delta <{\delta }_2$, then $x_i\in \mathrm{int}{M}_k$ for $N+1\le i\le n$. Moreover, by (4), we have $x_i\in \mathrm{int}(M_k\setminus M_{k-1})$ for $N+1\le i\le n$, and thus the claim is proved.

Finally, let us show that every pseudo-orbit ${\left\{x_i\right\}}_{i=0}^n\subset Y$ is shadowed by a true orbit of f. For ${ϵ}^{\prime }>0$, by the uniform continuity of f, we can choose $0<{\delta }^{\prime }<{\delta }_2$ such that every ${\delta }^{\prime }$-pseudo-orbit of f with length less than $N+1$ can be ${ϵ}^{\prime }$-shadowed by a true orbit of f. Thus, it is not difficult to show that any ${\delta }^{\prime }$-pseudo-orbit ${\left\{x_i\right\}}_{i=0}^n\subset Y$ of f can be shadowed by a true orbit of f reducing ${\delta }^{\prime }$ if necessary. Therefore, for the set Y, if we define $\tilde{m}\in \mathcal{M}\left(M\right)$ as

$$\tilde{m}\left(A\right)=\frac{m(A\cap Y)}{m\left(Y\right)}$$

for any Borel set $A\subset M$, then $\tilde{m}$ is shadowable, $m\left(Y\right)>1-ϵ$ and $f\left(Y\right)\subset Y$, and thus Theorem 1 is proved. □

We need a lemma to prove Theorem 2. Remark that if f possesses the filtrated pseudo-orbit shadowing property, then $f_{\left|\mathcal{R}\right(f)}$ has the shadowing property by definition. The following lemma proved in ([9], Proposition 2.3) will be used in the proof of the “only if” part of Theorem 2.

Lemma 1.

If $f_{\left|\mathcal{R}\right(f)}$ has the shadowing property, then the shadowing point can be taken from $\Omega \left(f\right)$ for any pseudo-orbit in $\Omega \left(f\right)$.

Proof of Theorem 2. 

To prove the if part of the theorem, suppose that f satisfies both Axiom A and the no-cycle condition. Then

$$\mathcal{R}\left(f\right)=\Omega \left(f\right)={\Lambda }_1\left(f\right)\cup \cdots \cup {\Lambda }_K\left(f\right)$$

for some $K\ge 1$, and there is a filtration $\varnothing =M_0\subset M_1\subset \cdots \subset M_K=M$ with respect to f. Here ${\Lambda }_k\left(f\right)$ is a hyperbolic basic set for $1\le k\le K$ (see [3,4]). To prove the filtrated pseudo-orbit shadowing property for f, we note that for any $1\le k\le K$ there is a neighborhood $U_k$ of ${\Lambda }_k\left(f\right)$ with the property that for all $ϵ>0$ there exists $\delta >0$ such that for any $\delta$-pseudo-orbit ${\left\{x_i\right\}}_{i=0}^n\subset U_k$$(n\in \mathbb{N})$ there exists $x\in M$ satisfying $d(f^i\left(x\right),x_i)<ϵ$ for all $0\le i\le n$ since ${\Lambda }_k\left(f\right)$ is hyperbolic (see [2,3]). By $(b.3)$, there is $m_k>0$ such that

$$\left(\bigcap _{m=-m_k}^{m_k}{f}^m(M_k\setminus M_{k-1})\right)\subset U_k.$$

Thus, we can see that there is ${\delta }_3>0$ such that for any $\delta$-pseudo-orbits ${\left\{x_i\right\}}_{i=0}^n\subset \mathrm{int}(M_k\setminus M_{k-1})$$(0<\delta <{\delta }_3,n\in \mathbb{N})$ of f, if $n\ge 2m_k+1$, then ${\left\{x_i\right\}}_{i=0}^n\subset U_k$. On the other hand, by the uniform continuity of f, every pseudo-orbit of f with length less than $2m_k+1$ can be shadowed by a true orbit of f. Therefore, it is not difficult to show that any $\delta$-pseudo-orbit ${\left\{x_i\right\}}_{i=0}^n\subset \mathrm{int}(M_k\setminus M_{k-1})$$(1\le k\le K)$ of f can be shadowed by a true orbit of f reducing $\delta$ if necessary.

Finally, it can be checked that since f is $\mathcal{R}$-stable (and thus, is $\Omega$-stable), any g$C^1$-nearby f also meets all of the conditions $\left(a\right)$, $\left(b\right)$ with respect to $\varnothing =M_0\subset M_1\subset \cdots \subset M_K=M$ (see [3], pp. 435–444) and has the filtrated pseudo-orbit shadowing property, and thus, the if part is proved.

To prove the only if part, let us denote by $\Omega \mathcal{S}$ the set of diffeomorphisms such that

-

$f:\Omega \left(f\right)\to \Omega \left(f\right)$ has the shadowing property; and

-

The shadowing point can be taken from $\Omega \left(f\right)$.

It was shown in ([10], Proposition 1) that any f in the $C^1$-interior of $\Omega \mathcal{S}$ satisfies both Axiom A and the no-cycle condition. Thus, to get the conclusion, it is enough to show that if $f\in \mathcal{FS}$, then f is in $\Omega \mathcal{S}$. Suppose that $f\in \mathcal{FS}$. Then $f_{\left|\mathcal{R}\right(f)}$ has the shadowing property, and thus, by Lemma 1, f is in $\Omega \mathcal{S}$ since the shadowing point can be taken from $\Omega \left(f\right)$. Thus, Theorem 2 is proved. □

Proof of Corollary 1. 

Let f be a quasi-Anosov diffeomorphism, so that f satisfies both Axiom A and the no-cycle condition. Since $f\in \mathrm{int}\mathcal{FS}$ by Theorem 2, every g$C^1$-nearby f admits an approximately shadowable Lebesgue measure by Theorem 1.

It is proved that every g$C^1$-nearby f is also quasi-Anosov by ([5], Lemma 1.6), and that f is Anosov if and only if $W^s\left(p\right)$ is the same dimension for all $p\in P\left(f\right)$ by ([5], Corollary 1). Suppose further that f is not Anosov. Then, since there are hyperbolic periodic points $p,q\in P\left(f\right)$ with different indices, that is, $dim{W}^s\left(p\right)\ne dim{W}^s\left(q\right)$, every g$C^1$-nearby f also has periodic points with different indices. Thus, g is quasi-Anosov but not Anosov by ([5], Corollary 1), so that g does not have the shadowing property by Theorem 3. □