Shadowing and the basins of terminal chain components

Abstract

We provide an alternative view of some results in [1, 3, 11]. In particular, we prove that (1) if a continuous self-map of a compact metric space has the shadowing, then the union of the basins of terminal chain components is a dense $G_\delta $-subset of the space; and (2) if a continuous self-map of a locally connected compact metric space has the shadowing, and if the chain recurrent set is totally disconnected, then the map is almost chain continuous.

1 Introduction

Shadowing is an important concept in the topological theory of dynamical systems (see [5, 18] for background). It was derived from the study of hyperbolic differentiable dynamics [4, 6] and generally refers to a situation in which coarse orbits, or pseudo-orbits, can be approximated by true orbits. Above all else, it is worth mentioning that the shadowing is known to be generic in the space of homeomorphisms or continuous self-maps of a closed differentiable manifold (see [19] and Theorem 1 of [16]) and so plays a significant role in the study of topologically generic dynamics.

Chain components are basic objects for global understanding of dynamical systems [9]. In this paper, we focus on attractor-like, or terminal, chain components and the basins of them. By a result (Corollary 6.16) of [11], if a continuous flow on a compact metric space has the so-called weak shadowing, then the union of the basins of terminal chain components is a dense $G_\delta $ -subset of the space. For any continuous self-map of a compact metric space, we strengthen it by assuming the standard shadowing (Theorem 1.1). Our proof is by a method related to but independent of a result (Proposition 22 in Section 7) of [1]. It is shown in [3] that topologically generic homeomorphisms of a closed differentiable manifold are almost chain continuous (see Introduction of [3] where the word “almost equicontinuous” is used). We also give an alternative proof of this fact by using the genericity of shadowing.

First, we define the chain components. Throughout, X denotes a compact metric space endowed with a metric d.

Definition 1.1 Given a continuous map $f\colon X\to X$ and $\delta>0$ , a finite sequence $(x_i)_{i=0}^{k}$ of points in X, where $k>0$ is a positive integer, is called a $\delta $ -chain of f if $d(f(x_i),x_{i+1})\le \delta $ for every $0\le i\le k-1$ . A $\delta $ -chain $(x_i)_{i=0}^{k}$ of f with $x_0=x_k$ is said to be a $\delta $ -cycle of f.

Let $f\colon X\to X$ be a continuous map. For any $x,y\in X$ and $\delta>0$ , the notation $x\rightarrow _\delta y$ means that there is a $\delta $ -chain $(x_i)_{i=0}^k$ of f with $x_0=x$ and $x_k=y$ . We write $x\rightarrow y$ if $x\rightarrow _\delta y$ for all $\delta>0$ . We say that $x\in X$ is a chain recurrent point for f if $x\rightarrow x$ , or equivalently, for every $\delta>0$ , there is a $\delta $ -cycle $(x_i)_{i=0}^{k}$ of f with $x_0=x_k=x$ . Let $CR(f)$ denote the set of chain recurrent points for f. We define a relation $\leftrightarrow $ in

$$\begin{align*}CR(f)^2=CR(f)\times CR(f) \end{align*}$$

by the following: for any $x,y\in CR(f)$ , $x\leftrightarrow y$ if and only if $x\rightarrow y$ and $y\rightarrow x$ . Note that $\leftrightarrow $ is a closed equivalence relation in $CR(f)^2$ and satisfies $x\leftrightarrow f(x)$ for all $x\in CR(f)$ . An equivalence class C of $\leftrightarrow $ is called a chain component for f. We regard the quotient space

$$\begin{align*}\mathcal{C}(f)=CR(f)/{\leftrightarrow} \end{align*}$$

as a space of chain components.

A subset S of X is said to be f-invariant if $f(S)\subset S$ . For an f-invariant subset S of X, we say that $f|_S\colon S\to S$ is chain transitive if for any $x,y\in S$ and $\delta>0$ , there is a $\delta $ -chain $(x_i)_{i=0}^k$ of $f|_S$ with $x_0=x$ and $x_k=y$ .

Remark 1.1 The following properties hold:

• • $CR(f)=\bigsqcup _{C\in \mathcal {C}(f)}C$ ,

• • every $C\in \mathcal {C}(f)$ is a closed f-invariant subset of $CR(f)$ ,

• • $f|_C\colon C\to C$ is chain transitive for all $C\in \mathcal {C}(f)$ ,

• • for any f-invariant subset S of X, if $f|_S\colon S\to S$ is chain transitive, then $S\subset C$ for some $C\in \mathcal {C}(f)$ .

Next, we recall the definition of terminal chain components. For $x\in X$ and a subset S of X, we denote by $d(x,S)$ the distance of x from S:

$$\begin{align*}d(x,S)=\inf_{y\in S}d(x,y). \end{align*}$$

Definition 1.2 We say that a closed f-invariant subset S of X is chain stable if for any $\epsilon>0$ , there is $\delta>0$ such that every $\delta $ -chain $(x_i)_{i=0}^k$ of f with $x_0\in S$ satisfies $d(x_i,S)\le \epsilon $ for all $0\le i\le k$ . Following [3], we say that $C\in \mathcal {C}(f)$ is terminal if C is chain stable. We denote by $\mathcal {C}_{\mathrm {ter}}(f)$ the set of terminal chain components for f.

Remark 1.2 For any continuous map $f\colon X\to X$ , a partial order $\le $ on $\mathcal {C}(f)$ is defined by the following: for all $C,D\in \mathcal {C}(f)$ , $C\le D$ if and only if $x\rightarrow y$ for some $x\in C$ and $y\in D$ . We can easily show that for any $C\in \mathcal {C}(f)$ , $C\in \mathcal {C}_{\mathrm {ter}}(f)$ if and only if C is maximal with respect to $\le $ ; that is, $C\le D$ implies $C=D$ for all $D\in \mathcal {C}(f)$ .

Given a continuous map $f\colon X\to X$ and $x\in X$ , the $\omega $ -limit set $\omega (x,f)$ of x for f is defined as the set of $y\in X$ such that

$$\begin{align*}\lim_{j\to\infty}f^{i_j}(x)=y \end{align*}$$

for some sequence $0\le i_1<i_2<\cdots $ . Note that $\omega (x,f)$ is a closed f-invariant subset of X and $f|_{\omega (x,f)}\colon \omega (x,f)\to \omega (x,f)$ is chain transitive. We denote by $C(x,f)$ the unique $C(x,f)\in \mathcal {C}(f)$ such that $\omega (x,f)\subset C(x,f)$ . For each $C\in \mathcal {C}(f)$ , we define the basin $W^s(C)$ of C by

$$\begin{align*}W^s(C)=\{x\in X\colon\lim_{i\to\infty}d(f^i(x),C)=0\}. \end{align*}$$

For every $x\in X$ , since

$$\begin{align*}\lim_{i\to\infty}d(f^i(x),\omega(x,f))=0, \end{align*}$$

we have $x\in W^s(C)$ if and only if $C=C(x,f)$ . This implies

$$\begin{align*}\{x\in X\colon C(x,f)\in\mathcal{C}_{\mathrm{ter}}(f)\}=\bigsqcup_{C\in\mathcal{C}_{\mathrm{ter}}(f)}W^s(C). \end{align*}$$

We also define the chain $\omega $ -limit set $\omega ^\ast (x,f)$ of x for f as the set of $y\in X$ such that for any $\delta>0$ and $N>0$ , there is a $\delta $ -chain $(x_i)_{i=0}^k$ of f with $x_0=x$ , $x_k=y$ , and $k\ge N$ . Note that $\omega ^\ast (x,f)$ is a closed f-invariant subset of X and chain stable. We have

$$\begin{align*}\omega(x,f)\subset C(x,f)\subset\omega^\ast(x,f). \end{align*}$$

Remark 1.3 The chain $\omega $ -limit set is denoted in [3] as $\omega \mathcal {C}(x,f)$ instead of $\omega ^\ast (x,f)$ .

The following lemma is obvious (see Section 1.4 of [3]).

Lemma 1.1 Let $f\colon X\to X$ be a continuous map.

1. (A) For any $x\in X$ , the following properties are equivalent:

   • – $C(x,f)\in \mathcal {C}_{\mathrm {ter}}(f)$ ,

   • – $\omega ^\ast (x,f)\subset C(x,f)$ ,

   • – $\omega ^\ast (x,f)=C(x,f)$ ,

   • – $f|_{\omega ^\ast (x,f)}\colon \omega ^\ast (x,f)\to \omega ^\ast (x,f)$ is chain transitive.

2. (B) For any $x\in X$ , the following properties are equivalent:

   • – $\omega (x,f)=C(x,f)=\omega ^\ast (x,f)$ ,

   • – $C(x,f)\in \mathcal {C}_{\mathrm {ter}}(f)$ and $\omega (x,f)=C(x,f)$ .

We give the definition of shadowing.

Definition 1.3 Let $f\colon X\to X$ be a continuous map and let $\xi =(x_i)_{i\ge 0}$ be a sequence of points in X. For $\delta>0$ , $\xi $ is called a $\delta $ -pseudo orbit of f if $d(f(x_i),x_{i+1})\le \delta $ for all $i\ge 0$ . For $\epsilon>0$ , $\xi $ is said to be $\epsilon $ -shadowed by $x\in X$ if $d(f^i(x),x_i)\leq \epsilon $ for all $i\ge 0$ . We say that f has the shadowing property if for any $\epsilon>0$ , there is $\delta>0$ such that every $\delta $ -pseudo orbit of f is $\epsilon $ -shadowed by some point of X.

For a topological space Z, a subset S of Z is called a $G_\delta $ -subset of Z if S is a countable intersection of open subsets of Z. If Z is completely metrizable, then by Baire Category Theorem, every countable intersection of open dense subsets of Z is dense in Z. We know that a subspace Y of a completely metrizable space Z is completely metrizable if and only if Y is a $G_\delta $ -subset of Z (see Theorem 24.12 of [20]).

For any continuous map $f\colon X\to X$ and $x\in X$ , let $\Omega (x,f)$ denote the set of $y\in X$ such that

$$\begin{align*}\lim_{j\to\infty}f^{i_j}(x_j)=y \end{align*}$$

for some sequence $0\le i_1<i_2<\cdots $ and $x_j\in X$ , $j\ge 1$ , with

$$\begin{align*}\lim_{j\to\infty}x_j=x. \end{align*}$$

Note that

$$\begin{align*}\omega(x,f)\subset\Omega(x,f)\subset\omega^\ast(x,f) \end{align*}$$

for all $x\in X$ . By Proposition 22 in Section 7 of [1], we know that

$$\begin{align*}\{x\in X\colon\omega(x,f)=\Omega(x,f)\} \end{align*}$$

is a dense $G_{\delta }$ -subset of X. The proof of this result in [1] is based on a nontrivial fact that the set of continuity points of a lower semicontinuous (lsc) set-valued map is a dense $G_{\delta }$ -subset. If f has the shadowing property, then we have

$$\begin{align*}\Omega(x,f)=\omega^\ast(x,f) \end{align*}$$

for all $x\in X$ . This can be proved as follows. Let $(\epsilon _j)_{j\ge 1}$ be a sequence of positive numbers with $\lim _{j\to \infty }\epsilon _j=0$ . Since f has the shadowing property, for each $j\ge 1$ , there is $\delta _j>0$ such that every $\delta _j$ -pseudo orbit of f is $\epsilon _j$ -shadowed by some point of X. Let $x\in X$ and $y\in \omega ^\ast (x,f)$ . Since $y\in \omega ^\ast (x,f)$ , we have a sequence $(x_i^{(j)})_{i=0}^{k_j}$ , $j\ge 1$ , of $\delta _j$ -chains of f with $x_0^{(j)}=x$ , $x_{k_j}^{(j)}=y$ , and $k_j< k_{j+1}$ for all $j\ge 1$ . By the choice of $\delta _j$ , we obtain $x_j\in X$ , $j\ge 1$ , such that $d(x_j,x)=d(x_j,x_0^{(j)})\le \epsilon _j$ and $d(f^{k_j}(x_j),y)=d(f^{k_j}(x_j),x_{k_j}^{(j)})\le \epsilon _j$ for all $j\ge 1$ . It follows that $0<k_1<k_2<\cdots $ ,

$$\begin{align*}\lim_{j\to\infty}x_j=x, \end{align*}$$

and

$$\begin{align*}\lim_{j\to\infty}f^{k_j}(x_j)=y. \end{align*}$$

Thus, $y\in \Omega (x,f)$ . Since $x\in X$ and $y\in \omega ^\ast (x,f)$ are arbitrary, we conclude that

$$\begin{align*}\omega^\ast(x,f)\subset\Omega(x,f) \end{align*}$$

for all $x\in X$ , completing the proof. It follows that if a continuous map $f\colon X\to X$ has the shadowing property, then

$$\begin{align*}\{x\in X\colon\omega(x,f)=\Omega(x,f)=\omega^\ast(x,f)\} \end{align*}$$

is a dense $G_{\delta }$ -subset of X; therefore,

$$\begin{align*}\{x\in X\colon\omega(x,f)=C(x,f)=\omega^\ast(x,f)\}=\{x\in X\colon {C(x,f)\in\mathcal{C}_{\mathrm{ter}}(f) \text{ and } \omega(x,f)=C(x,f)}\} \end{align*}$$

is a dense $G_{\delta }$ -subset of X (see [11] and [17] for related results). The main aim of this paper is to give an alternative proof of the following statement.

Theorem 1.1 If a continuous map $f\colon X\to X$ has the shadowing property, then

$$\begin{align*}V(f)=\{x\in X\colon C(x,f)\in\mathcal{C}_{\mathrm{ter}}(f)\} \end{align*}$$

and

$$\begin{align*}W(f)=\{x\in V(f)\colon\omega(x,f)=C(x,f)\} \end{align*}$$

are dense $G_\delta $ -subsets of X.

Given a continuous map $f\colon X\to X$ and $x\in X$ , we say that f is chain continuous at x if for any $\epsilon>0$ , there is $\delta>0$ such that every $\delta $ -pseudo orbit $(x_i)_{i\ge 0}$ of f with $x_0=x$ is $\epsilon $ -shadowed by x [2]. We denote by $CC(f)$ the set of chain continuity points for f. The notion of chain continuity is closely related to odometers. An odometer (or an adding machine) is defined as follows. Let $m=(m_j)_{j\ge 1}$ be an increasing sequence of positive integers with $m_j|m_{j+1}$ for all $j\ge 1$ . Let $X_j$ , $j\ge 1$ , denote the quotient group $\mathbb {Z}/m_j\mathbb {Z}$ with the discrete topology. Let $\pi _j\colon X_{j+1}\to X_j$ , $j\ge 1$ , be the natural projections and let

$$\begin{align*}X_m=\{x=(x_j)_{j\ge1}\in\prod_{j\ge1}X_j\colon\pi_j(x_{j+1})=x_j \text{ for all } j\ge 1\}. \end{align*}$$

As a closed subspace of $\prod _{j\ge 1}X_j$ with the product topology, $X_m$ is a compact metrizable space. Consider the map $g_m\colon X_m\to X_m$ defined by

$$\begin{align*}g_m(x)_j=x_j+1 \end{align*}$$

for all $x=(x_j)_{j\ge 1}\in X_m$ and $j\ge 1$ . Note that $g_m$ is a homeomorphism. We say that $(X_m,g_m)$ is an odometer with the periodic structure m. We say that a closed f-invariant subset S of X is an odometer if $(S,f|_S)$ is topologically conjugate to an odometer. This is equivalent to that S is a Cantor space and

$$\begin{align*}f|_{S}\colon S\to S \end{align*}$$

is a minimal equicontinuous homeomorphism (see Theorem 4.4 of [15]). By Theorem 7.5 of [3], we know that for any $x\in X$ , $x\in CC(f)$ if and only if

$$\begin{align*}\omega(x,f)=C(x,f)=\omega^\ast(x,f) \end{align*}$$

and $C(x,f)$ is a periodic orbit or an odometer. By Lemma 1.1, this is equivalent to that $C(x,f)\in \mathcal {C}_{\mathrm {ter}}(f)$ and $C(x,f)$ is a periodic orbit or an odometer. We say that X is locally connected if for any $x\in X$ and any open subset U of X with $x\in U$ , we have $x\in V\subset U$ for some open connected subset V of X. A subspace S of X is said to be totally disconnected if every connected component of S is a singleton. If X is locally connected and $CR(f)$ is totally disconnected, then due to Theorem 5.1 of [8] or Theorem B of [10], every $C\in \mathcal {C}_{\mathrm {ter}}(f)$ is a periodic orbit or an odometer. By these facts, we obtain the following lemma.

Lemma 1.2 Let $f\colon X\to X$ be a continuous map. If X is locally connected and $CR(f)$ is totally disconnected, then for any $x\in X$ , the following properties are equivalent:

• • $x\in CC(f)$ ,

• • $\omega (x,f)=C(x,f)=\omega ^\ast (x,f)$ ,

• • $C(x,f)\in \mathcal {C}_{\mathrm {ter}}(f)$ .

Let $f\colon X\to X$ be a continuous map. For any $j,l\ge 1$ , let $C_{j,l}$ denote the set of $x\in X$ such that there is a neighborhood U of x for which every $\frac {1}{j}$ -pseudo orbit $(x_i)_{i\ge 0}$ of f with $x_0\in U$ is $\frac {1}{l}$ -shadowed by $x_0$ . We see that $C_{j,l}$ is an open subset of X for all $j,l\ge 1$ and

$$\begin{align*}CC(f)=\bigcap_{l\ge1}\bigcup_{j\ge1}C_{j,l}. \end{align*}$$

Thus, $CC(f)$ is a $G_{\delta }$ -subset of X. We say that f is almost chain continuous if $CC(f)$ is a dense $G_\delta $ -subset of X. By Theorem 1.1 and Lemma 1.2, we obtain the following theorem.

Theorem 1.2 Let $f\colon X\to X$ be a continuous map. If X is locally connected, f has the shadowing property, and if $CR(f)$ is totally disconnected, then f is almost chain continuous.

We present a corollary of Theorem 1.2. For a closed differentiable manifold M, let $\mathcal {H}(M)$ (resp. $\mathcal {C}(M)$ ) denote the set of homeomorphisms (resp. continuous self-maps) of M, endowed with the $C^0$ -topology. It is shown in [3] that generic $f\in \mathcal {H}(M)$ (resp. $f\in \mathcal {C}(M)$ , if $\dim {M}>1$ ) is almost chain continuous (see Introduction of [3] where the word “almost equicontinuous” is used). Note that the shadowing is generic in $\mathcal {H}(M)$ [19] and also generic in $\mathcal {C}(M)$ [16, Theorem 1]. Moreover, by results of [3, 14], we know that for generic $f\in \mathcal {H}(M)$ (resp. $f\in \mathcal {C}(M)$ ), $CR(f)$ is totally disconnected (see Introduction of [3] and Theorem 3.3 of [14]). Thus, by Theorem 1.2, we obtain the following corollary.

Corollary 1.1 Generic $f\in \mathcal {H}(M)$ (resp. $f\in \mathcal {C}(M)$ ) is almost chain continuous.

Our results also apply to the case where X is not a manifold. We say that X is a dendrite if X is connected, locally connected, and contains no simple closed curves. The shadowing is proved to be generic in the space of continuous self-maps of a dendrite (see [7] and [13, Theorem 19]). However, by Corollary 5.2 of [14], a generic continuous self-map of a dendrite has the totally disconnected chain recurrent set. By Theorem 1.2, we conclude that a generic continuous self-map of a dendrite is almost chain continuous.

This paper consists of two sections. In the next section, we prove Theorem 1.1.

2 Proof of Theorem 1.1

In this section, we prove Theorem 1.1. The proof is based on the following lemma in [12].

Lemma 2.1 [12, Lemma 2.1]

For any continuous map $f\colon X\to X$ and $x\in X$ , there is $C\in \mathcal {C}_{\mathrm {ter}}(f)$ such that for every $\delta>0$ , there is a $\delta $ -chain $(x_i)_{i=0}^k$ of f with $x_0=x$ and $x_k\in C$ .

We need one more lemma. In what follows, for $x\in X$ and a subset S of X, we denote by $d(x,S)$ the distance of x from S:

$$\begin{align*}d(x,S)=\inf_{y\in S}d(x,y). \end{align*}$$

We also denote by $U_r(S)$ , $r>0$ , the open r-neighborhood of S:

$$\begin{align*}U_r(S)=\{x\in X\colon d(x,S)<r\}. \end{align*}$$

Lemma 2.2 For any continuous map $f\colon X\to X$ and $x\in X$ , if $C(x,f)\in \mathcal {C}_{\mathrm {ter}}(f)$ , then $C(\cdot ,f)\colon X\to \mathcal {C}(f)$ is continuous at x.

Proof Let $x\in X$ and $C=C(x,f)$ . If $C\in \mathcal {C}_{\mathrm {ter}}(f)$ (i.e., C is chain stable), then for any $\epsilon>0$ , we have $\delta>0$ such that every $\delta $ -chain $(x_i)_{i=0}^k$ of f with $d(x_0,C)\le \delta $ satisfies $d(x_i,C)\le \epsilon /2$ for all $0\le i\le k$ . It follows that $d(y,C)\le \delta $ implies

$$\begin{align*}\omega^\ast(y,f)\subset U_\epsilon(C) \end{align*}$$

for all $y\in X$ . Since

$$\begin{align*}\lim_{i\to\infty}d(f^i(x),C)=0, \end{align*}$$

we have $d(f^i(x),C)\le \delta /2$ for some $i\ge 0$ . By taking $\gamma>0$ such that $d(x,z)\le \gamma $ implies $d(f^i(x),f^i(z))\le \delta /2$ for all $z\in X$ , we obtain $d(f^i(z),C)\le \delta $ and so

$$\begin{align*}C(z,f)\subset\omega^\ast(z,f)=\omega^\ast(f^i(z),f)\subset U_\epsilon(C) \end{align*}$$

for all $z\in X$ with $d(x,z)\le \gamma $ . Since $\epsilon>0$ is arbitrary, this implies that $C(\cdot ,f)\colon X\to \mathcal {C}(f)$ is continuous at x, completing the proof.

By using these lemmas, we prove Theorem 1.1.

Proof of Theorem 1.1

First, we show that $V(f)$ is a dense $G_\delta $ -subset of X. Fix a sequence $(\epsilon _j)_{j\ge 1}$ of positive numbers such that $\epsilon _1>\epsilon _2>\cdots $ and

$$\begin{align*}\lim_{j\to\infty}\epsilon_j=0. \end{align*}$$

For any $j\ge 1$ and $C\in \mathcal {C}_{\mathrm {ter}}(f)$ , we take $\delta _{j,C}>0$ such that $x\in U_{\delta _{j,C}}(C)$ implies

$$\begin{align*}\omega^\ast(x,f)\subset U_{\epsilon_j}(C) \end{align*}$$

for all $x\in X$ . Let

$$\begin{align*}U_{j,C}=U_{\delta_{j,C}}(C) \end{align*}$$

for all $j\ge 1$ and $C\in \mathcal {C}_{\mathrm {ter}}(f)$ . We define a subset V of X by

$$\begin{align*}V=\bigcap_{j\ge1}\bigcup_{C\in\mathcal{C}_{\mathrm{ter}}(f)}\bigcup_{m\ge0}f^{-m}(U_{j,C}). \end{align*}$$

Note that V is a $G_{\delta }$ -subset of X. Since f has the shadowing property, by Lemma 2.1, we see that for every $x\in X$ , there is $C\in \mathcal {C}_{\mathrm {ter}}(f)$ such that

$$\begin{align*}x\in\overline{\bigcup_{m\ge0}f^{-m}(U_{j,C})} \end{align*}$$

for all $j\ge 1$ . This can be proved as follows. For $x\in X$ , fix $C\in \mathcal {C}_{\mathrm {ter}}(f)$ as in Lemma 2.1 and $\gamma _l>0$ , $l\ge 1$ , with $\lim _{l\to \infty }\gamma _l=0$ . There are $\beta _l>0$ , $l\ge 1$ , and a sequence $(x_i^{(l)})_{i=0}^{k_l}$ , $l\ge 1$ , of $\beta _l$ -chains of f such that for each $l\ge 1$ ,

• • every $\beta _l$ -pseudo orbit of f is $\gamma _l$ -shadowed by some point of X,

• • $x^{(l)}_0=x$ and $x_{k_l}^{(l)}\in C$ .

By taking $x_l\in X$ , $l\ge 1$ , with $d(x_l,x)=d(x_l,x^{(l)}_0)\le \gamma _l$ and $d(f^{k_l}(x_l),C)\le d(f^{k_l}(x_l),x_{k_l}^{(l)})\le \gamma _l$ , we obtain $\lim _{l\to \infty }x_l=x$ and

$$\begin{align*}x_l\in f^{-k_l}(U_{j,C})\subset\bigcup_{m\ge0}f^{-m}(U_{j,C}) \end{align*}$$

for any fixed $j\ge 1$ and all sufficiently large $l\ge 1$ , implying

$$\begin{align*}x\in\overline{\bigcup_{m\ge0}f^{-m}(U_{j,C})} \end{align*}$$

for all $j\ge 1$ . This proves the claim. It follows that

$$\begin{align*}X\subset\bigcup_{C\in\mathcal{C}_{\mathrm{ter}}(f)}\bigcap_{j\ge1}\overline{\bigcup_{m\ge0}f^{-m}(U_{j,C})}\subset\bigcup_{C\in\mathcal{C}_{\mathrm{ter}}(f)}\overline{\bigcup_{m\ge0}f^{-m}(U_{j,C})}\subset\overline{\bigcup_{C\in\mathcal{C}_{\mathrm{ter}}(f)}\bigcup_{m\ge0}f^{-m}(U_{j,C})} \end{align*}$$

for all $j\ge 1$ . With the aid of Baire Category Theorem, this implies that V is a dense $G_\delta $ -subset of X. It remains to prove that $V(f)=V$ . Given any $x\in V(f)$ , by $C(x,f)\in \mathcal {C}_{\mathrm {ter}}(f)$ and

$$\begin{align*}x\in\bigcap_{j\ge1}\bigcup_{m\ge0}f^{-m}(U_{j,C(x,f)})\subset V, \end{align*}$$

we have $x\in V$ . It follows that $V(f)\subset V$ . Conversely, let $x\in V$ . For each $j\ge 1$ , we take $C_j\in \mathcal {C}_{\mathrm {ter}}(f)$ and $m_j\ge 0$ such that

$$\begin{align*}x\in f^{-m_j}(U_{j,C_j}). \end{align*}$$

Then, because $\mathcal {C}(f)=CR(f)/{\leftrightarrow }$ is a compact metrizable space, there are a sequence $1\le j_1<j_2<\cdots $ and $C\in \mathcal {C}(f)$ such that

$$\begin{align*}\lim_{l\to\infty}C_{j_l}=C \end{align*}$$

in $\mathcal {C}(f)$ . Note that for every $\epsilon>0$ , we have

$$\begin{align*}C_{j_l}\subset U_\epsilon(C) \end{align*}$$

for all sufficiently large $l\ge 1$ . For every $l\ge 1$ , by

$$\begin{align*}f^{m_{j_l}}(x)\in U_{{j_l},C_{j_l}}, \end{align*}$$

we have

$$\begin{align*}\omega^\ast(x,f)=\omega^\ast(f^{m_{j_l}}(x),f)\subset U_{\epsilon_{j_l}}(C_{j_l}). \end{align*}$$

By

$$\begin{align*}\lim_{l\to\infty}\epsilon_{j_l}=0, \end{align*}$$

we obtain

$$\begin{align*}\omega^\ast(x,f)\subset U_{2\epsilon}(C) \end{align*}$$

for all $\epsilon>0$ ; thus, $\omega ^\ast (x,f)\subset C$ . From Lemma 1.1, it follows that $C=C(x,f)\in \mathcal {C}_{\mathrm {ter}}(f)$ , implying $x\in V(f)$ . Since $x\in V$ is arbitrary, we conclude that $V\subset V(f)$ , proving the claim.

Next, we show that $W(f)$ is a dense $G_\delta $ -subset of X. Since $V(f)$ is a dense $G_\delta $ -subset of X, it suffices to show that $W(f)$ is a dense $G_\delta $ -subset of $V(f)$ . Letting

$$\begin{align*}W=\bigcap_{j\ge1}\bigcap_{m\ge0}\{x\in V(f)\colon C(x,f)\subset U_{\frac{1}{j}}(\{f^i(x)\colon i\ge m\})\}, \end{align*}$$

we have $W=W(f)$ . Let

$$\begin{align*}W_{j,m}=\{x\in V(f)\colon C(x,f)\subset U_{\frac{1}{j}}(\{f^i(x)\colon i\ge m\})\} \end{align*}$$

for all $j\ge 1$ and $m\ge 0$ . Given any $x\in W_{j,m}$ , $j\ge 1$ , $m\ge 0$ , by compactness of $C(x,f)$ , there are $0<r<\frac {1}{j}$ and $n\ge m$ such that

$$\begin{align*}C(x,f)\subset U_r(\{f^i(x)\colon m\le i\le n\}). \end{align*}$$

We take $\epsilon>0$ with $r+2\epsilon <\frac {1}{j}$ . Since $x\in V(f)$ and so $C(x,f)\in \mathcal {C}_{\mathrm {ter}}(f)$ , by Lemma 2.2, there is $a>0$ such that $d(x,y)<a$ implies

$$\begin{align*}C(y,f)\subset U_\epsilon(C(x,f)) \end{align*}$$

for all $y\in X$ . By continuity of f, we have $b>0$ such that $d(x,y)<b$ implies

$$\begin{align*}\{f^i(x)\colon m\le i\le n\}\subset U_\epsilon(\{f^i(y)\colon m\le i\le n\}) \end{align*}$$

for all $y\in X$ . It follows that $d(x,y)<\min \{a,b\}$ implies

$$\begin{align*}C(y,f)\subset U_{r+2\epsilon}(\{f^i(y)\colon m\le i\le n\})\subset U_{\frac{1}{j}}(\{f^i(y)\colon m\le i\le n\})\subset U_{\frac{1}{j}}(\{f^i(y)\colon i\ge m\}) \end{align*}$$

for all $y\in X$ . Since $x\in W_{j,m}$ is arbitrary, $W_{j,m}$ is an open subset of $V(f)$ . Since $j\ge 1$ and $m\ge 0$ are arbitrary, we conclude that W is a $G_\delta $ -subset of $V(f)$ . It remains to prove that W is a dense subset of $V(f)$ . Let $j\ge 1$ and $m\ge 0$ . Given any $x\in V(f)$ and $\epsilon>0$ , since $C(x,f)\in \mathcal {C}_{\mathrm {ter}}(f)$ , by Lemma 2.2, there is $0<a<\epsilon /2$ such that $d(x,y)<2a$ implies

$$\begin{align*}C(y,f)\subset U_{\frac{1}{3j}}(C(x,f)) \end{align*}$$

for all $y\in X$ . Since f has the shadowing property, we see that

$$\begin{align*}C(x,f)\subset U_{\frac{1}{3j}}(\{f^i(p)\colon i\ge m\}) \end{align*}$$

for some $p\in X$ with $d(x,p)<a$ . By compactness of $C(x,f)$ , we obtain

$$\begin{align*}C(x,f)\subset U_{\frac{1}{3j}}(\{f^i(p)\colon m\le i\le n\}) \end{align*}$$

for some $n\ge m$ . By continuity of f, we have $b>0$ such that $d(p,q)<b$ implies

$$\begin{align*}\{f^i(p)\colon m\le i\le n\}\subset U_{\frac{1}{3j}}(\{f^i(q)\colon m\le i\le n\}) \end{align*}$$

for all $q\in X$ . Since $V(f)$ is a dense subset of X, we have $d(p,q)<\min \{a,b\}$ for some $q\in V(f)$ . Note that

$$\begin{align*}d(x,q)\le d(x,p)+d(p,q)<2a<\epsilon. \end{align*}$$

It follows that

$$\begin{align*}C(q,f)\subset U_{\frac{1}{3j}}(C(x,f))\subset U_{\frac{1}{j}}(\{f^i(q)\colon m\le i\le n\})\subset U_{\frac{1}{j}}(\{f^i(q)\colon i\ge m\}), \end{align*}$$

implying $q\in W_{j,m}$ . Since $x\in V(f)$ and $\epsilon>0$ are arbitrary, $W_{j,m}$ is an open dense subset of $V(f)$ . Since $j\ge 1$ and $m\ge 0$ are arbitrary, we conclude that W is a dense subset of $V(f)$ , proving the claim. Thus, the theorem has been proved.

We conclude with a remark on the proof.

Remark 2.1

• • The proof shows that $V(f)$ and $W(f)$ are $G_\delta $ -subsets of X for every continuous map $f\colon X\to X$ .

• • For any continuous map $f\colon X\to X$ , we can show that if f has the shadowing property, then

  $$\begin{align*}V(f)=\{x\in X\colon C(\cdot,f)\colon X\to\mathcal{C}(f) \text{ is continuous at } x\}. \end{align*}$$
  By this, since $\mathcal {C}(f)$ is a compact metrizable space, we can show that $V(f)$ is a $G_\delta $ -subset of X.

• • Let $f\colon X\to X$ be a continuous map and let $\xi =(x_i)_{i\ge 0}$ be a sequence of points in X. For $\delta>0$ , $\xi $ is called a $\delta $ -limit-pseudo orbit of f if $d(f(x_i),x_{i+1})\le \delta $ for all $i\ge 0$ , and

  $$\begin{align*}\lim_{i\to\infty}d(f(x_i),x_{i+1})=0. \end{align*}$$
  For $\epsilon>0$ , $\xi $ is said to be $\epsilon $ -limit shadowed by $x\in X$ if $d(f^i(x),x_i)\leq \epsilon $ for all $i\ge 0$ , and
  $$\begin{align*}\lim_{i\to\infty}d(f^i(x),x_i)=0. \end{align*}$$
  We say that f has the s-limit shadowing property if for any $\epsilon>0$ , there is $\delta>0$ such that every $\delta $ -limit-pseudo orbit of f is $\epsilon $ -limit shadowed by some point of X. When f has the s-limit shadowing property, by Lemma 2.1, we can easily show that $W(f)$ is a dense subset of X.