1 Introduction

Shadowing is an important concept in the topological theory of dynamical systems (see [Reference Aoki and Hiraide5, Reference Pilyugin18] for background). It was derived from the study of hyperbolic differentiable dynamics [Reference Anosov4, Reference Bowen6] 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 [Reference Pilyugin and Plamenevskaya19] and Theorem 1 of [Reference Mazur and Oprocha16]) and so plays a significant role in the study of topologically generic dynamics.

Chain components are basic objects for global understanding of dynamical systems [Reference Conley9]. In this paper, we focus on attractor-like, or terminal, chain components and the basins of them. By a result (Corollary 6.16) of [Reference Hurley11], 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 [Reference Akin1]. It is shown in [Reference Akin, Hurley and Kennedy3] that topologically generic homeomorphisms of a closed differentiable manifold are almost chain continuous (see Introduction of [Reference Akin, Hurley and Kennedy3] 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.

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 $x_k=y$ .

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:

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

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

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

The following lemma is obvious (see Section 1.4 of [Reference Akin, Hurley and Kennedy3]).

We give the definition of shadowing.

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 [Reference Willard20]).

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 [Reference Akin1], 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 [Reference Akin1] 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 [Reference Hurley11] and [Reference Pilyugin17] 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 [Reference Akin2]. 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 [Reference Kůrka15]). By Theorem 7.5 of [Reference Akin, Hurley and Kennedy3], 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 [Reference Buescu and Stewart8] or Theorem B of [Reference Hirsch and Hurley10], every $C\in \mathcal {C}_{\mathrm {ter}}(f)$ is a periodic orbit or an odometer. By these facts, we obtain the following lemma.

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.

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 [Reference Akin, Hurley and Kennedy3] that generic $f\in \mathcal {H}(M)$ (resp. $f\in \mathcal {C}(M)$ , if $\dim {M}>1$ ) is almost chain continuous (see Introduction of [Reference Akin, Hurley and Kennedy3] where the word “almost equicontinuous” is used). Note that the shadowing is generic in $\mathcal {H}(M)$ [Reference Pilyugin and Plamenevskaya19] and also generic in $\mathcal {C}(M)$ [Reference Mazur and Oprocha16, Theorem 1]. Moreover, by results of [Reference Akin, Hurley and Kennedy3, Reference Krupski, Omiljanowski and Ungeheuer14], we know that for generic $f\in \mathcal {H}(M)$ (resp. $f\in \mathcal {C}(M)$ ), $CR(f)$ is totally disconnected (see Introduction of [Reference Akin, Hurley and Kennedy3] and Theorem 3.3 of [Reference Krupski, Omiljanowski and Ungeheuer14]). Thus, by Theorem 1.2, we obtain the following corollary.

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 [Reference Brian, Meddaugh and Raines7] and [Reference Kościelniak, Mazur, Oprocha and Kubica13, Theorem 19]). However, by Corollary 5.2 of [Reference Krupski, Omiljanowski and Ungeheuer14], 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.