Some Results on Shadowing and Local Entropy Properties of Dynamical Systems

Abstract

We consider some local entropy properties of dynamical systems under the assumption of shadowing. In the first part, we give necessary and sufficient conditions for shadowable points to be certain entropy points. In the second part, we give some necessary and sufficient conditions for (non) h-expansiveness under the assumption of shadowing and chain transitivity; and use the result to present a counter-example for a question raised by Artigue et al. (Proc Am Math Soc 150:3369–3378, 2022).

1 Introduction

Shadowing, introduced by Anosov (1967) and Bowen (1975), is a feature of hyperbolic dynamical systems and has played an important role in the global theory of dynamical systems [see Aoki and Hiraide (1994) or Pilyugin (1999) for background]. It generally refers to a phenomenon in which coarse orbits, or pseudo-orbits, are approximated by true orbits. In Morales (2016), by splitting the global shadowing into pointwise shadowings, Morales introduced the notion of shadowable points, which gives a tool for a local description of the shadowing phenomena. The study of shadowable points has been extended to shadowable points for flows (Aponte and Villavicencio 2018); pointwise stability and persistence (Das et al. 2020; Das and Khan 2022; Dong et al. 2019; Jung et al. 2020; Koo et al. 2018); shadowable measures (Shin 2019); average shadowable and specification points (Das et al. 2019; Das and Khan 2021); eventually shadowable points (Dong et al. 2020); shadowable points of set-valued dynamical systems (Luo et al. 2020); and so on.

In Kawaguchi (2017b), some sufficient conditions are given for a shadowable point to be an entropy point. Recall that the notion of entropy points is obtained by a concentration of positive topological entropy at a point (Ye and Zhang 2007). Also, in Arbieto and Rego (2020), the notion of shadowable points is applied to obtain pointwise sufficient conditions for positive topological entropy (see also Arbieto and Rego 2022). In the first part of this paper, we improve the result of Kawaguchi (2017b) by giving necessary and sufficient conditions for shadowable points to be certain entropy points. The h-expansiveness is another local entropy property of dynamical systems (Bowen 1972). In the second part of this paper, we present several necessary and sufficient conditions for (non) h-expansiveness under the assumption of shadowing and chain transitivity; and use the result to obtain a counter-example for a question in Artigue et al. (2022).

We begin with a definition. Throughout, X denotes a compact metric space endowed with a metric d.

Definition 1.1

Given a continuous map $f:X\rightarrow X$ 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 $f:X\rightarrow X$ 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 any $\delta >0$, 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

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

by: 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 denote by ${\mathcal {C}}(f)$ the set of chain components for f.

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 $f|_S:S\rightarrow S$ 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),

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

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

Let $f:X\rightarrow X$ be a continuous map. For $x\in X$, we define a subset C(x) of X by

$$\begin{aligned} C(x)=\{x\}\cup \{y\in X:x\rightarrow y\}. \end{aligned}$$

By this definition, we easily see that for any $x\in X$, C(x) is a closed f-invariant subset of X. We say that a closed f-invariant subset S of X is chain stable if for any $\epsilon >0$, there is $\delta >0$ for which every $\delta$-chain $(x_i)_{i=0}^k$ of f with $x_0\in S$ satisfies $d(x_i,S)\le \epsilon$ for all $0\le i\le k$. A proof of the following lemma is given in Sect. 3.

Lemma 1.1

C(x) is chain stable for all $x\in X$.

Remark 1.2

For any $x\in X$, since C(x) is chain stable, it satisfies the following properties

• $CR(f|_{C(x)})=C(x)\cap CR(f)$,

• for every $C\in {\mathcal {C}}(f)$, $C\subset C(x)$ if and only if $C\cap C(x)\ne \emptyset$,

• ${\mathcal {C}}(f|_{C(x)})=\{C\in {\mathcal {C}}(f):C\subset C(x)\}$.

Let $f:X\rightarrow X$ 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 $d(f^i(x),x_i)\le \epsilon$ for all $i\ge 0$.

Definition 1.2

Given a continuous map $f:X\rightarrow X$, $x\in X$ is called a shadowable point for f 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 some $y\in X$. We denote by Sh(f) the set of shadowable points for f.

For a continuous map $f:X\rightarrow X$ and a subset S of X, we say that f has the shadowing on S if for any $\epsilon >0$, there is $\delta >0$ such that every $\delta$-pseudo orbit $(x_i)_{i\ge 0}$ of f with $x_i\in S$ for all $i\ge 0$ is $\epsilon$-shadowed by some $y\in X$. We say that f has the shadowing property if f has the shadowing on X.

The next lemma is a basis for the formulation of Theorems 1.1 and 1.2.

Lemma 1.2

For a continuous map $f:X\rightarrow X$ and $x\in X$, the following conditions are equivalent

• (1) $x\in Sh(f)$,

• (2) $C(x)\subset Sh(f)$,

• (3) f has the shadowing on C(x).

Next, we recall the definition of entropy points from Ye and Zhang (2007). Let $f:X\rightarrow X$ be a continuous map. For $n\ge 1$, the metric $d_n$ on X is defined by

$$\begin{aligned} d_n(x,y)=\max _{0\le i\le n-1}d(f^i(x),f^i(y)) \end{aligned}$$

for all $x,y\in X$. For $n\ge 1$ and $r>0$, a subset E of X is said to be (n, r)-separated if $d_n(x,y)>r$ for all $x,y\in E$ with $x\ne y$. Let K be a subset of X. For $n\ge 1$ and $r>0$, let $s_n(f,K,r)$ denote the largest cardinality of an (n, r)-separated subset of K. We define h(f, K, r) and h(f, K) by

$$\begin{aligned} h(f,K,r)=\limsup _{n\rightarrow \infty }\frac{1}{n}\log {s_n(f,K,r)} \end{aligned}$$

and

$$\begin{aligned} h(f,K)=\lim _{r\rightarrow 0}h(f,K,r). \end{aligned}$$

We also define the topological entropy $h_{\textrm{top}}(f)$ of f by $h_{\textrm{top}}(f)=h(f,X)$.

Definition 1.3

Let $f:X\rightarrow X$ be a continuous map. For $x\in X$, we denote by ${\mathcal {K}}(x)$ the set of closed neighborhoods of x.

• (1) Ent(f) is the set of $x\in X$ such that $h(f,K)>0$ for all $K\in {\mathcal {K}}(x)$,

• (2) For $r>0$, $Ent_r(f)$ is the set of $x\in X$ such that $h(f,K,r)>0$ for all $K\in {\mathcal {K}}(x)$,

• (3) For $r>0$ and $b>0$, $Ent_{r,b}(f)$ is the set of $x\in X$ such that $h(f,K,r)\ge b$ for all $K\in {\mathcal {K}}(x)$.

Remark 1.3

The following properties hold

• Ent(f), $Ent_r(f)$, $r>0$, and $Ent_{r,b}(f)$, $r,b>0$, are closed f-invariant subsets of X,

• $$\begin{aligned} Ent(f)\subset Ent_r(f)\subset Ent_{r,b}(f) \end{aligned}$$

  for all $r,b>0$,

• for any closed subset K of X and $r>0$, if $h(f,K,r)>0$, then $K\cap Ent_r(f)\ne \emptyset$,

• for any closed subset K of X and $r,b>0$, if $h(f,K,r)\ge b$, then $K\cap Ent_{r,b}(f)\ne \emptyset$.

For a continuous map $f:X\rightarrow X$, we define ${\mathcal {C}}_o(f)$ as the set of $C\in {\mathcal {C}}(f)$ such that C is a periodic orbit or an odometer. We also define ${\mathcal {C}}_{no}(f)$ to be

$$\begin{aligned} {\mathcal {C}}_{no}(f)={\mathcal {C}}(f)\setminus {\mathcal {C}}_o(f). \end{aligned}$$

We refer to Sect. 2.1 for various matters related to this definition.

The first theorem characterizes the shadowable points that are entropy points of a certain type.

Theorem 1.1

Given a continuous map $f:X\rightarrow X$ and $x\in Sh(f)$,

$$\begin{aligned} x\in \bigcup _{r>0}Ent_r(f) \end{aligned}$$

if and only if ${\mathcal {C}}_{no}(f|_{C(x)})\ne \emptyset$.

The following theorem characterizes the shadowable points that are entropy points of other type.

Theorem 1.2

Given a continuous map $f:X\rightarrow X$ and $x\in Sh(f)$,

$$\begin{aligned} x\in \bigcup _{r,b>0}Ent_{r,b}(f) \end{aligned}$$

if and only if $h_{\textrm{top}}(f|_{C(x)})=h(f,C(x))>0$.

Remark 1.4

In Ye and Zhang (2007), a point of

$$\begin{aligned} \bigcup _{r,b>0}Ent_{r,b}(f) \end{aligned}$$

is called a uniform entropy point for f.

Remark 1.5

In Sect. 3, we give an example of a continuous map $f:X\rightarrow X$ such that

• f has the shadowing property and so satisfies $X=Sh(f)$,

• $$\begin{aligned} Ent(f)\setminus \bigcup _{r>0}Ent_r(f) \end{aligned}$$

  is a non-empty set.

Before we state the next theorem, we introduce some definitions.

Definition 1.4

For a continuous map $f:X\rightarrow X$ and $(x,y)\in X^2$,

• (x, y) is called a distal pair for f if

  $$\begin{aligned} \liminf _{i\rightarrow \infty }d(f^i(x),f^i(y))>0, \end{aligned}$$

• (x, y) is called a proximal pair for f if

  $$\begin{aligned} \liminf _{i\rightarrow \infty }d(f^i(x),f^i(y))=0, \end{aligned}$$

• (x, y) is called an asymptotic pair for f if

  $$\begin{aligned} \limsup _{i\rightarrow \infty }d(f^i(x),f^i(y))=0. \end{aligned}$$

• (x, y) is called a scrambled pair for f if

  $$\begin{aligned} \limsup _{i\rightarrow \infty }d(f^i(x),f^i(y))>0\,\text { and }\,\liminf _{i\rightarrow \infty }d(f^i(x),f^i(y))=0. \end{aligned}$$

For a continuous map $f:X\rightarrow X$ and $x\in X$, the $\omega$-limit set $\omega (x,f)$ of x for f is defined to be the set of $y\in X$ such that $\lim _{j\rightarrow \infty }f^{i_j}(x)=y$ for some sequence $0\le i_1<i_2<\cdots$. Note that $\omega (x,f)$ is a closed f-invariant subset of X and satisfies $y\rightarrow z$ for all $y,z\in \omega (x,f)$. For every $x\in X$, we have $\omega (x,f)\subset C$ for some $C\in {\mathcal {C}}(f)$ and such C satisfies $C\subset C(x)$.

Remark 1.6

Let $f:X\rightarrow X$ be a continuous map.

• For a closed f-invariant subset S of X and $e>0$, we say that $x\in S$ is an e-sensitive point for $f|_S:S\rightarrow S$ if for any $\epsilon >0$, there is $y\in S$ such that $d(x,y)\le \epsilon$ and $d(f^i(x),f^i(y))>e$ for some $i\ge 0$. We define $Sen_e(f|_S)$ to be the set of e-sensitive points for $f|_S$ and

  $$\begin{aligned} Sen(f|_S)=\bigcup _{e>0}Sen_e(f|_S). \end{aligned}$$

• A closed f-invariant subset M of X is said to be a minimal set for f if closed f-invariant subsets of M are only $\emptyset$ and M. This is equivalent to $M=\omega (x,f)$ for all $x\in M$.

In Kawaguchi (2017b), the author gave three sufficient conditions for a shadowable point to be an entropy point. The next theorem refines Corollary 1.1 of Kawaguchi (2017b).

Theorem 1.3

Let $f:X\rightarrow X$ be a continuous map. For any $x\in X$ and $C\in {\mathcal {C}}(f)$ with $\omega (x,f)\subset C$, if one of the following conditions is satisfied, then $C\in {\mathcal {C}}_{no}(f|_{C(x)})$.

• (1) $\omega (x,f)\cap Sen(f|_{CR(f)})\ne \emptyset$,

• (2) there is $y\in X$ such that $(x,y)\in X^2$ is a scrambled pair for f,

• (3) $\omega (x,f)$ is not a minimal set for f.

Remark 1.7

For a continuous map $f:X\rightarrow X$, $y\in X$ is called a minimal point for f if $y\in \omega (y,f)$ and $\omega (y,f)$ is a minimal set for f. Due to Theorem 8.7 of Furstenberg (1981), we know that for any $x\in X$, there is a minimal point $y\in X$ for f such that (x, y) is a proximal pair for f. If $\omega (x,f)$ is not a minimal set for f, then it follows that (x, y) is a scrambled pair for f, thus (3) always implies (2).

We consider another local property of dynamical systems so-called h-expansiveness Bowen (1972). Let $f:X\rightarrow X$ be a continuous map. For $x\in X$ and $\epsilon >0$, let

$$\begin{aligned} \Phi _{\epsilon }(x)=\{y\in X:d(f^i(x),f^i(y))\le \epsilon \,\,\text {for all }i\ge 0\} \end{aligned}$$

and

$$\begin{aligned} h_f^*(\epsilon )=\sup _{x\in X}h(f,\Phi _{\epsilon }(x)). \end{aligned}$$

We say that f is h-expansive if $h_f^*(\epsilon )=0$ for some $\epsilon >0$. The following theorem gives several conditions equivalent to (non) h-expansiveness under the assumption of shadowing and chain transitivity.

Theorem 1.4

Let $f:X\rightarrow X$ be a continuous map. If f is chain transitive and has the shadowing property, then the following conditions are equivalent

• (1) f is not h-expansive,

• (2) for any $\epsilon >0$, there is $r>0$ such that for every $\delta >0$, there is a pair

  $$\begin{aligned} ((x_i)_{i=0}^k,(y_i)_{i=0}^k) \end{aligned}$$

  of $\delta$-chains of f with $(x_0,x_k)=(y_0,y_k)$ and

  $$\begin{aligned} r\le \max _{0\le i\le k}d(x_i,y_i)\le \epsilon , \end{aligned}$$

• (3) for any $\epsilon >0$, there are $m\ge 1$ and a closed $f^m$-invariant subset Y of X such that

  $$\begin{aligned} \sup _{i\ge 0}d(f^i(x),f^i(y))\le \epsilon \end{aligned}$$

  for all $x,y\in Y$ and there is a factor map

  $$\begin{aligned} \pi :(Y,f^m)\rightarrow (\{0,1\}^{\mathbb {N}},\sigma ), \end{aligned}$$

  where $\sigma :\{0,1\}^{\mathbb {N}}\rightarrow \{0,1\}^{\mathbb {N}}$ is the shift map.

• (4) for any $\epsilon >0$, there is a scrambled pair $(x,y)\in X^2$ for f such that

  $$\begin{aligned} \sup _{i\ge 0}d(f^i(x),f^i(y))\le \epsilon . \end{aligned}$$

In Sect. 4, we use this theorem to obtain a counter-example for a question in Artigue et al. (2022). We shall make some definitions to precisely state the properties that are satisfied by the example.

Definition 1.5

Let $f:X\rightarrow X$ 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

$$\begin{aligned} \lim _{i\rightarrow \infty }d(f(x_i),x_{i+1})=0. \end{aligned}$$

For $\epsilon >0$, $\xi$ is said to be $\epsilon$-limit shadowed by $x\in X$ if $d(f^i(x),x_i)\le \epsilon$ for all $i\ge 0$, and

$$\begin{aligned} \lim _{i\rightarrow \infty }d(f^i(x),x_i)=0. \end{aligned}$$

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.

Remark 1.8

If f has the s-limit shadowing property, then f satisfies the shadowing property.

Definition 1.6

Let $f:X\rightarrow X$ be a homeomorphism. For $x\in X$ and $\epsilon >0$, let

$$\begin{aligned} \Gamma _\epsilon (x)=\{y\in X:d(f^i(x),f^i(y))\le \epsilon \,\,\text {for all }i\in {\mathbb {Z}}\}. \end{aligned}$$

We say that f is

• expansive if there is $e>0$ such that $\Gamma _e(x)=\{x\}$ for all $x\in X$,

• countably-expansive if there is $e>0$ such that $\Gamma _e(x)$ is a countable set for all $x\in X$,

• cw-expansive if there is $e>0$ such that $\Gamma _e(x)$ is totally disconnected for all $x\in X$.

A continuous map $f:X\rightarrow X$ is said to be transitive (resp. mixing) if for any non-empty open subsets U, V of X, it holds that $f^j(U)\cap V\ne \emptyset$ for some $j>0$ (resp. for all $j\ge i$ for some $i>0$).

Remark 1.9

If f is transitive, then f is chain transitive, and the converse holds when f has the shadowing property.

In Sect. 4, by using Theorem 1.4, we give an example of a homeomorphism $f:X\rightarrow X$ (Example 4.1) such that

• (1) X is totally disconnected and so f is cw-expansive,

• (2) f is mixing,

• (3) f is h-expansive,

• (4) f has the s-limit shadowing property,

• (5) f is not countably-expansive,

• (6) f satisfies $X_e=\emptyset$, where

  $$\begin{aligned} X_e=\{x\in X:\Gamma _\epsilon (x)=\{x\}\,\text { for some }\epsilon >0\}. \end{aligned}$$

In Artigue et al. (2020), it is proved that if a homeomorphism $f:X\rightarrow X$ has the L-shadowing property, that is, a kind of two-sided s-limit shadowing property, then

$$\begin{aligned} f|_{CR(f)}:CR(f)\rightarrow CR(f) \end{aligned}$$

is expansive if and only if $f|_{CR(f)}$ is countably-expansive if and only if $f|_{CR(f)}$ is h-expansive (see Corollary C of Artigue et al. (2020)). Example 4.1 shows that even if a homeomorphism $f:X\rightarrow X$ satisfies the s-limit shadowing property, this equivalence does not hold. The example also gives a negative answer to the following question in Artigue et al. (2022) (see Question 3 of Artigue et al. (2022)):

Question

Is $X_e$ non-empty for every transitive h-expansive and cw-expansive homeomorphism $f:X\rightarrow X$ satisfying the shadowing property?

This paper consists of four sections. In Sect. 2, we collect some definitions, notations, and facts that are used in this paper. In Sect. 3, we prove Lemmas 1.1 and 1.2; prove Theorems 1.1, 1.2, and 1.3; and give an example mentioned in Remark 1.5. In Sect. 4, we prove Theorem 1.4 and give an example mentioned above (Example 4.1) after proving some auxiliary lemmas.

2 Preliminaries

In this section, we briefly collect some definitions, notations, and facts that are used in this paper.

2.1 Odometers, Equicontinuity, and Chain Continuity

An odometer (also called an adding machine) is defined as follows. Given an increasing sequence $m=(m_k)_{k\ge 1}$ of positive integers such that $m_1\ge 1$ and $m_k$ divides $m_{k+1}$ for each $k=1,2,\dots$, we define

• $X(k)=\{0,1,\dots ,m_k-1\}$ (with the discrete topology),

• $$\begin{aligned} X_m=\left\{ (x_k)_{k\ge 1}\in \prod _{k\ge 1}X(k):x_k\equiv x_{k+1}\pmod {m_k}\,\text { for all }k\ge 1\right\} , \end{aligned}$$

• $g_m(x)_k=x_k+1\pmod {m_k}$ for all $x=(x_k)_{k\ge 1}\in X_m$ and $k\ge 1$.

We regard $X_m$ as a subspace of the product space $\prod _{k\ge 1}X(k)$. The homeomorphism

$$\begin{aligned} g_m:X_m\rightarrow X_m \end{aligned}$$

(or $(X_m,g_m)$) is called an odometer with the periodic structure m.

Let $f:X\rightarrow X$ be a continuous map and let S be a closed f-invariant subset of X. We say that $f|_S:S\rightarrow S$ is

• equicontinuous if for every $\epsilon >0$, there is $\delta >0$ such that any $x,y\in S$ with $d(x,y)\le \delta$ satisfies

  $$\begin{aligned} \sup _{i\ge 0}d(f^i(x),f^i(y))\le \epsilon , \end{aligned}$$

• chain continuous if for every $\epsilon >0$, there is $\delta >0$ such that any $\delta$-pseudo orbits $(x_i)_{i\ge 0}$ and $(y_i)_{i\ge 0}$ of $f|_S$ with $x_0=y_0$ satisfies

  $$\begin{aligned} \sup _{i\ge 0}d(x_i,y_i)\le \epsilon . \end{aligned}$$

Recall that for a continuous map $f:X\rightarrow X$, ${\mathcal {C}}_o(f)$ is defined as the set of $C\in {\mathcal {C}}(f)$ such that C is a periodic orbit or an odometer, that is, $(C,f|_C)$ is topologically conjugate to an odometer.

Lemma 2.1

For a continuous map $f:X\rightarrow X$, the following conditions are equivalent

• (1) ${\mathcal {C}}(f)={\mathcal {C}}_o(f)$,

• (2) $f|_{CR(f)}:CR(f)\rightarrow CR(f)$ is an equicontinuous homeomorphism and CR(f) is totally disconnected,

• (3) $f|_{CR(f)}:CR(f)\rightarrow CR(f)$ is chain continuous.

Proof

We prove the implication $(1)\implies (2)$. Since ${\mathcal {C}}(f)={\mathcal {C}}_o(f)$,

• (A) every $C\in {\mathcal {C}}(f)$ is totally disconnected,

• (B) $f|_{CR(f)}$ is a distal homeomorphism, that is, every $(x,y)\in CR(f)^2$ is a distal pair for $f|_{CR(f)}$.

Since the quotient space

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

is totally disconnected, by (A), we obtain that CR(f) is totally disconnected. By (B) and Corollary 1.9 of Auslander et al. (2007), we conclude that $f|_{CR(f)}$ is an equicontinuous homeomorphism. For a proof of $(2)\implies (3)$ (resp. $(3)\implies (1)$), we refer to Lemma 3.3 (resp. Section 6) of Kawaguchi (2021a). $\square$

By applying Lemma 2.1 to $f|_C:C\rightarrow C$, $C\in {\mathcal {C}}(f)$, we obtain the following corollary.

Corollary 2.1

For a continuous map $f:X\rightarrow X$ and $C\in {\mathcal {C}}(f)$, the following conditions are equivalent

• (1) $C\in {\mathcal {C}}_o(f)$,

• (2) $f|_C:C\rightarrow C$ is an equicontinuous homeomorphism and C is totally disconnected,

• (3) $f|_C:C\rightarrow C$ is chain continuous.

2.2 Factor Maps and Inverse Limit

For two continuous maps $f:X\rightarrow X$, $g:Y\rightarrow Y$, where X, Y are compact metric spaces, a continuous map $\pi :X\rightarrow Y$ is said to be a factor map if $\pi$ is surjective and satisfies $\pi \circ f=g\circ \pi$. A factor map $\pi :X\rightarrow Y$ is also denoted as

$$\begin{aligned} \pi :(X,f)\rightarrow (Y,g). \end{aligned}$$

Given an inverse sequence of factor maps

$$\begin{aligned} \pi =(\pi _n:(X_{n+1},f_{n+1})\rightarrow (X_n,f_n))_{n\ge 1}, \end{aligned}$$

let

$$\begin{aligned} X=\{x=(x_n)_{n\ge 1}\in \prod _{n\ge 1}X_n:\pi _n(x_{n+1})=x_n\,\text { for all }n\ge 1\}, \end{aligned}$$

which is a compact metric space. Then, a continuous map $f:X\rightarrow X$ is well-defined by $f(x)=(f_n(x_n))_{n\ge 1}$ for all $x=(x_n)_{n\ge 1}\in X$. We call

$$\begin{aligned} (X,f)=\lim _\pi (X_n,f_n) \end{aligned}$$

the inverse limit system. It is easy to see that f is transitive (resp. mixing) if and only if $f_n:X_n\rightarrow X_n$ is transitive (resp. mixing) for all $n\ge 1$. It is also easy to see that f has the shadowing property if $f_n:X_n\rightarrow X_n$ has the shadowing property for all $n\ge 1$.

3 Proofs of Theorems 1.1, 1.2, and 1.3

In this section, we prove Lemmas 1.1 and 1.2; prove Theorems 1.1, 1.2, and 1.3; and give an example mentioned in Remark 1.5.

First, we prove Lemma 1.1.

Proof of Lemma 1.1

If C(x) is not chain stable, then there is $r>0$ such that for any $\delta >0$, there is a $\delta$-chain $x^{(\delta )}=(x_i^{(\delta )})_{i=0}^{k_\delta }$ of f with $x_0^{(\delta )}\in C(x)$ and

$$\begin{aligned} d(x_{k_\delta }^{(\delta )}, C(x))\ge r. \end{aligned}$$

Then, there are a sequence $0<\delta _1>\delta _2>\cdots$ and $y,z\in X$ such that the following conditions are satisfied

• $\lim _{j\rightarrow \infty }\delta _j=0$,

• $\lim _{j\rightarrow \infty }x_0^{(\delta _j)}=y$ and $\lim _{j\rightarrow \infty }x_{k_{\delta _j}}^{(\delta _j)}=z$.

It follows that $y\in C(x)$, $d(z,C(x))\ge r>0$ and so $z\not \in C(x)$; and $y\rightarrow z$. However, if $y=x$, we obtain $x\rightarrow z$ implying $z\in C(x)$, a contradiction. If $y\ne x$, by $x\rightarrow y$ and $y\rightarrow z$, we obtain $x\rightarrow z$ implying $z\in C(x)$, a contradiction. Thus, the lemma has been proved. $\square$

Next, we prove Lemma 1.2.

Proof of Lemma 1.2

We prove the implication $(1)\implies (2)$. Let $x\in Sh(f)$ and $y\in C(x)\setminus \{x\}$. For any $\epsilon >0$, since $x\in Sh(f)$, 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 some $z\in X$. Since $y\in C(x)\setminus \{x\}$ and so $x\rightarrow y$, we have a $\delta$-chain $\alpha =(y_i)_{i=0}^k$ of f with $y_0=x$ and $y_k=y$. For any $\delta$-pseudo orbit $\beta =(z_i)_{i\ge 0}$ of f with $z_0=y$, we consider a $\delta$-pseudo orbit

$$\begin{aligned} \xi =\alpha \beta =(x_i)_{i\ge 0}=(y_0,y_1,\dots ,y_{k-1},z_0,z_1,z_2,\dots ) \end{aligned}$$

of f. Then, since $x_0=y_0=x$, $\xi$ is $\epsilon$-shadowed by some $z\in X$ and so $\beta$ is $\epsilon$-shadowed by $f^k(z)$. Since $\epsilon >0$ is arbitrary, we obtain $y\in Sh(f)$, thus $(1)\implies (2)$ has been proved.

Next, we prove the implication $(2)\implies (3)$. For a closed subset K of X, if $K\subset Sh(f)$, then by Lemma 2.4 of Kawaguchi (2017a), for any $\epsilon >0$, there is $\delta >0$ such that every $\delta$-pseudo orbit $(x_i)_{i\ge 0}$ of f with $x_0\in K$ is $\epsilon$-shadowed by some $y\in X$. Since C(x) is a closed subset of X, this clearly implies that if $C(x)\subset Sh(f)$, then f has the shadowing on C(x).

Finally, we prove the implication $(3)\implies (1)$. If f has the shadowing on C(x), then for any $\epsilon >0$, there is $\delta >0$ such that every $\delta$-pseudo orbit $(x_i)_{i\ge 0}$ of f with $x_i\in C(x)$ for all $i\ge 0$ is $\epsilon$-shadowed by some $y\in X$. Since $x\in C(x)$ and C(x) is chain stable, if $\gamma >0$ is sufficiently small, then for every $\gamma$-pseudo orbit $\xi =(y_i)_{i\ge 0}$ of f with $y_0=x$, by taking $x_i\in C(x)$, $i>0$, with $d(y_i,C(x))=d(y_i,x_i)$ for all $i>0$, we have that

• $d(x_i,y_i)\le \epsilon$ for each $i>0$,

• $$\begin{aligned} (x_i)_{i\ge 0}=(x,x_1,x_2,x_3,\dots ) \end{aligned}$$

  is a $\delta$-pseudo orbit of f with $x_i\in C(x)$ for all $i\ge 0$ and so is $\epsilon$-shadowed by some $y\in X$.

It follows that $\xi$ is $2\epsilon$-shadowed by y. Since $\epsilon >0$ is arbitrary, we obtain $x\in Sh(f)$, thus $(3)\implies (1)$ has been proved. This completes the proof of Lemma 1.2. $\square$

We give a proof of Theorem 1.1.

Proof of Theorem 1.1

First, we prove the “if” part. Let $C\in {\mathcal {C}}_{no}(f|_{C(x)})$. Due to Corollary 2.1, since $f|_C:C\rightarrow C$ is not chain continuous, there are $p\in C$ and $e>0$ such that for any $\delta >0$, there are $\delta$-chains $(x_i)_{i=0}^k$ and $(y_i)_{i=0}^k$ of $f|_C$ with $x_0=y_0=p$ and $d(x_k,y_k)>e$. Fix $0<r<e$ and take any $\epsilon >0$ with $r+2\epsilon <e$. Since $x\in Sh(f)$, there is $\delta _0>0$ such that every $\delta _0$-pseudo orbit $(x_i)_{i\ge 0}$ of f with $x_0=x$ is $\epsilon$-shadowed by some $y\in X$. We fix a pair

$$\begin{aligned} ((x_i)_{i=0}^K,(y_i)_{i=0}^K) \end{aligned}$$

of $\delta _0$-chains $f|_C$ with $x_0=y_0=p$ and $d(x_K,y_K)>e$. Since $C\subset C(x)$, we have $x\rightarrow q$ for some $q\in C$. We also fix a $\delta _0$-chain $\alpha =(z_i)_{i=0}^L$ of f with $z_0=x$ and $z_L=q$. Since $f|_C$ is chain transitive, by compactness of C, there is $M>0$ such that for any $w\in C$, there is a $\delta _0$-chain $(w_i)_{i=0}^m$ of $f|_C$ with $w_0=w$, $w_m=p$, and $m\le M$. It follows that for any $w\in C$, there is a pair

$$\begin{aligned} (a^w,b^w)=((a_i^w)_{i=0}^{k_w},(b_i^w)_{i=0}^{k_w}) \end{aligned}$$

of $\delta _0$-chains of $f|_C$ with $a_0^w=b_0^w=w$, $d(a_{k_w}^w,b_{k_w}^w)>e$, and $k_w\le K+M$. Given any $N\ge 1$ and $s=(s_i)_{i=1}^N\in \{a,b\}^N$, we inductively define a family of $\delta _0$-chains

$$\begin{aligned} \alpha (s,n)=(c(s,n)_i)_{i=0}^{k(s,n)} \end{aligned}$$

of $f|_C$, $1\le n\le N$, by $\alpha (s,1)=s_1^q$ and $\alpha (s,n+1)=s_{n+1}^{c(s,n)_{k(s,n)}}$ for any $1\le n\le N-1$. Then, we consider a family of $\delta _0$-chains

$$\begin{aligned} \alpha (s)=(c(s)_i)_{i=0}^{k(s)}=\alpha \alpha (s,1)\alpha (s,2)\cdots \alpha (s,N) \end{aligned}$$

of f, $s\in \{a,b\}^N$. Note that $c(s)_0=x$ and $k(s)\le L+N(K+M)$ for all $s\in \{a,b\}^N$; and for any $s,t\in \{a,b\}^N$ with $s\ne t$, we have $d(c(s)_i,c(t)_i)>e$ for some

$$\begin{aligned} 0\le i \le \min \{k(s),k(t)\}\le L+N(K+M). \end{aligned}$$

By the choice of $\delta _0$, for every $s\in \{a,b\}^N$, there is $x(s)\in X$ such that $d(f^i(x(s)),c(s)_i)\le \epsilon$ for all $0\le i\le k(s)$. It follows that

$$\begin{aligned} \{x(s):s\in \{a,b\}^N\} \end{aligned}$$

is an $(L+N(K+M),r)$-separated subset of $B_\epsilon (x)=\{y\in X:d(x,y)\le \epsilon \}$. Since $N\ge 1$ is arbitrary, we obtain

$$\begin{aligned} h(f,B_\epsilon (x),r)&=\limsup _{n\rightarrow \infty }\frac{1}{n}\log {s_n(f,B_\epsilon (x),r)}\\&\ge \limsup _{N\rightarrow \infty }\frac{1}{L+N(K+M)}\log {s_{L+N(K+M)}(f,B_\epsilon (x),r)}\\&\ge \limsup _{N\rightarrow \infty }\frac{1}{L+N(K+M)}\log {2^N}\\&=\frac{1}{K+M}\log {2}>0. \end{aligned}$$

Since $\epsilon >0$ with $r+2\epsilon <e$ is arbitrary, we conclude that $x\in Ent_r(f)$, proving the “if” part.

Next, we prove the “only if” part. Let $x\in Ent_r(f)$ for some $r>0$. Due to Lemma 2.1, it suffices to show that

$$\begin{aligned} f|_{CR(f|_{C(x)})}:CR(f|_{C(x)})\rightarrow CR(f|_{C(x)}) \end{aligned}$$

is not chain continuous. For any $\epsilon >0$, let

$$\begin{aligned} S_\epsilon =\{y\in C(x):d(y,CR(f|_{C(x)}))\le \epsilon \} \end{aligned}$$

and

$$\begin{aligned} T_\epsilon =\{y\in C(x):d(y,CR(f|_{C(x)}))\ge \epsilon \}. \end{aligned}$$

Since

$$\begin{aligned} CR(f|_{C(x)})= C(x)\cap CR(f), \end{aligned}$$

we have $T_\epsilon \cap CR(f)=\emptyset$; therefore, for any $p\in T_\epsilon$, we can take a neighborhood $U_p$ of p in X such that

• (1) $d(a,b)\le r$ and $d(f(a),f(b))\le \epsilon$ for all $a,b\in U_p$,

• (2) $f^i(c)\not \in U_p$ for all $c\in U_p$ and $i>0$.

We take $p_1,p_2,\dots ,p_M\in T_\epsilon$ with $T_\epsilon \subset \bigcup _{j=1}^M U_{p_j}$. Let $U=\bigcup _{j=1}^M U_{p_j}$and take $0<\Delta \le \epsilon$ such that

$$\begin{aligned} \{z\in X:d(z,T_\epsilon )\le \Delta \}\subset U. \end{aligned}$$

Since $x\in C(x)$ and C(x) is chain stable, we can take a closed neighborhood K of x in X such that

• (3) $d(a,b)\le \epsilon$ for all $a,b\in K$,

• (4) $d(f^i(c),C(x))\le \Delta$ for all $c\in K$ and $i\ge 0$.

For any $q\in X$ and $n\ge 1$, let

$$\begin{aligned} A(q,n)=\{0\le i\le n-1:f^i(q)\in U\} \end{aligned}$$

and take

$$\begin{aligned} g(q,n):A(q,n)\rightarrow \{U_{p_j}:1\le j\le M\} \end{aligned}$$

such that $f^i(q)\in g(q,n)(i)$ for every $i\in A(q,n)$. By (2), we have $|A(q,n)|\le M$ for all $q\in X$ and $n\ge 1$. Note that

$$\begin{aligned} |\{(A(q,n),g(q,n)):q\in X\}|\le \sum _{k=0}^{\min \{n,M\}}\left( {\begin{array}{c}n\\ k\end{array}}\right) M^k\le (M+1)n^M M^M. \end{aligned}$$

for all $n\ge 1$. Since $x\in Ent_r(f)$, we have

$$\begin{aligned} h(f,K,r)=\limsup _{n\rightarrow \infty }\frac{1}{n}\log {s_n(f,K,r)}>0; \end{aligned}$$

therefore,

$$\begin{aligned} s_n(f,K,r)>(M+1)n^M M^M \end{aligned}$$

for some $n\ge 1$. This implies that there are $u,v\in K$ such that $d_n(u,v)>r$ and

$$\begin{aligned} (A(u,n),g(u,n))=(A(v,n),g(v,n)). \end{aligned}$$

We fix $0\le N\le n-1$ with $d(f^N(u),f^N(v))>r$ and let

$$\begin{aligned} (A,g)=(A(u,n),g(u,n))=(A(v,n),g(v,n)). \end{aligned}$$

If

$$\begin{aligned} A\cap \{0\le l\le N\}=\emptyset , \end{aligned}$$

then $f^l(u),f^l(v)\not \in U$ for all $0\le l\le N$. By (4) and the choice of $\Delta$, for any $0\le l\le N$, we obtain

$$\begin{aligned} \{f^l(u),f^l(v)\}\subset \{w\in X:d(w,S_\epsilon )\le \Delta \}. \end{aligned}$$

It follows that

$$\begin{aligned} \max \{d(f^l(u),CR(f|_{C(x)})),d(f^l(v),CR(f|_{C(x)}))\}\le \epsilon +\Delta \le 2\epsilon \end{aligned}$$

for all $0\le l\le N$. Moreover, by $u,v\in K$ and (3), we obtain $d(u,v)\le \epsilon$. If

$$\begin{aligned} A\cap \{0\le l\le N\}\ne \emptyset , \end{aligned}$$

letting

$$\begin{aligned} L=\max \,[A\cap \{0\le l\le N\}], \end{aligned}$$

we have $f^L(u),f^L(v)\in g(L)$ and $g(L)\in \{U_{p_j}:1\le j\le M\}$. By (1), we have $L<N$ and $d(f^{L+1}(u),f^{L+1}(v))\le \epsilon$. By

$$\begin{aligned} A\cap \{L+1\le l\le N\}=\emptyset , \end{aligned}$$

(4), and the choice of $\Delta$, similarly as above, we obtain

$$\begin{aligned} \max \{d(f^l(u),CR(f|_{C(x)})),d(f^l(v),CR(f|_{C(x)}))\}\le \epsilon +\Delta \le 2\epsilon \end{aligned}$$

for all $L+1\le l\le N$. Since $\epsilon >0$ is arbitrary, we conclude that $f|_{CR(f|_{C(x)})}$ is not chain continuous, thus the “only if” part has been proved. This completes the proof of Theorem 1.1. $\square$

For the proof of Theorem 1.2, we need two lemmas.

Lemma 3.1

Let $f:X\rightarrow X$ be a continuous map.

• (1) For any $x,y\in X$ and $r>0$, if $x\in Sh(f)$ and $y\in C(x)\cap Ent_r(f)$, then $x\in Ent_s(f)$ for all $0<s<r$.

• (2) For any $x,y\in X$ and $r,b>0$, if $x\in Sh(f)$ and $y\in C(x)\cap Ent_{r,b}(f)$, then $x\in Ent_{s,b}(f)$ for all $0<s<r$.

Proof

Let $x\in Sh(f)$ and $y\in C(x){\setminus }\{x\}$. For any $0<s<r$, we fix $\epsilon >0$ with $s+2\epsilon <r$. Since $x\in Sh(f)$, there is $\delta >0$ such that every $\delta$-pseudo orbit of $(x_i)_{i\ge 0}$ of f with $x_0=x$ is $\epsilon$-shadowed by some $z\in X$. Since $y\in C(x)\setminus \{x\}$ and so $x\rightarrow y$, we have a $\delta /2$-chain $(y_i)_{i=0}^k$ of f with $y_0=x$ and $y_k=y$. For $K\in {\mathcal {K}}(y)$, $n\ge 1$, and $r>0$, we take an (n, r)-separated subset E(K, n, r) of K with $|E(K,n,r)|=s_n(f,K,r)$. If K is sufficiently small, then for any $p\in E(K,n,r)$,

$$\begin{aligned} (z_i^p)_{i=0}^{k+n-1}=(y_0,y_1,\dots ,y_{k-1},p,f(p),\dots ,f^{n-1}(p)) \end{aligned}$$

is a $\delta$-chain of f with $z_0^p=y_0=x$ and so there is $z_p\in X$ with $d(f^i(z_p),z_i^p)\le \epsilon$ for all $0\le i\le k+n-1$. It follows that

$$\begin{aligned} \{z_p:p\in E(K,n,r)\} \end{aligned}$$

is a $(k+n,s)$-separated subset of $B_\epsilon (x)=\{w\in X:d(x,w)\le \epsilon \}$ and so

$$\begin{aligned} s_{k+n}(f,B_\epsilon (x),s)\ge |E(K,n,r)|=s_n(f,K,r), \end{aligned}$$

implying

$$\begin{aligned} h(f,B_\epsilon (x),s)&=\limsup _{n\rightarrow \infty }\frac{1}{k+n}\log {s_{k+n}(f,B_\epsilon (x),s)}\\&\ge \limsup _{n\rightarrow \infty }\frac{1}{k+n}\log {s_n(f,K,r)}\\&=h(f,K,r). \end{aligned}$$

Since $\epsilon >0$ with $s+2\epsilon <r$ is arbitrary, if $y\in Ent_r(f)$ (resp. $y\in Ent_{r,b}(f)$ for some $b>0$), we obtain $x\in Ent_s(f)$ (resp. $x\in Ent_{s,b}(f)$). Since $0<s<r$ is arbitrary, the lemma has been proved. $\square$

Let $f:X\rightarrow X$ be a continuous map. For $\delta ,r>0$ and $n\ge 1$, we say that two $\delta$-chains $(x_i)_{i=0}^n$ and $(y_i)_{i=0}^n$ of f is (n, r)-separated if $d(x_i,y_i)>r$ for some $0\le i\le n$. Let

$$\begin{aligned} s_n(f,X,r,\delta ) \end{aligned}$$

denote the largest cardinality of a set of (n, r)-separated $\delta$-chains of f. The following lemma is from Misiurewicz (1986).

Lemma 3.2

(Misiurewicz)

$$\begin{aligned} h_{\textrm{top}}(f)=\lim _{r\rightarrow 0}\lim _{\delta \rightarrow 0}\limsup _{n\rightarrow \infty }\frac{1}{n}\log {s_n(f,X,r,\delta )}. \end{aligned}$$

We give a proof of Theorem 1.2.

Proof of Theorem 1.2

First, we prove the “if” part. Since

$$\begin{aligned} h_{\textrm{top}}(f|_{C(x)})=h(f,C(x))>0, \end{aligned}$$

we have $h(f,C(x),r)>0$ for some $r>0$. Taking $0<b\le h(f,C(x),r)$, we obtain $C(x)\cap Ent_{r,b}(f)\ne \emptyset$. Since $x\in Sh(f)$, by Lemma 3.1, this implies $x\in Ent_{s,b}(f)$ for all $0<s<r$, thus the “if” part has been proved.

Next, we prove the“only if” part. Let $x\in Ent_{r_0,b}(f)$ for some $r_0,b>0$. Then, we have

$$\begin{aligned} h(f,K,r_0)=\limsup _{n\rightarrow \infty }\frac{1}{n}\log {s_n(f,K,r_0)}\ge b \end{aligned}$$

for all $K\in {\mathcal {K}}(x)$. Since C(x) is chain stable, taking $0<s<r_0$, we obtain

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{1}{n}\log {s_n(f,C(x),s,\delta )}\ge b \end{aligned}$$

for all $\delta >0$. From Lemma 3.2, it follows that

$$\begin{aligned} h_{\textrm{top}}(f|_{C_{(x)}})&=\lim _{r\rightarrow 0}\lim _{\delta \rightarrow 0}\limsup _{n\rightarrow \infty }\frac{1}{n}\log {s_n(f,C(x),r,\delta )}\\&\ge \lim _{\delta \rightarrow 0}\limsup _{n\rightarrow \infty }\frac{1}{n}\log {s_n(f,C(x),s,\delta )}\\&\ge b>0, \end{aligned}$$

thus the “only if” part has been proved. This completes the proof of Theorem 1.2. $\square$

Next, we prove Theorem 1.3. The proof of the following lemma is left to the reader.

Lemma 3.3

Let $f:X\rightarrow X$ be a continuous map and let $C\in {\mathcal {C}}(f)$. For any $\epsilon >0$, there is $\delta >0$ such that every $\delta$-chain $(x_i)_{i=0}^k$ of $f|_{CR(f)}$ with $x_0\in C$ satisfies $d(x_i,C)\le \epsilon$ for all $0\le i\le k$.

Proof of Theorem 1.3

Due to Corollary 2.1, it is sufficient to show that each of the three conditions implies that $f|_C:C\rightarrow C$ is not chain continuous.

• (1) Taking $y\in \omega (x,f)\cap Sen(f|_{CR(f)})$, we have

  $$\begin{aligned} y\in C\cap Sen_{e_0}(f|_{CR(f)}) \end{aligned}$$

  for some $e_0>0$. Taking $0<e<e_0$, we obtain that for any $\delta >0$, there are $\delta$-chains $(x_i)_{i=0}^k$ and $(y_i)_{i=0}^k$ of $f|_{CR(f)}$ with $x_0=y_0=x$ and $d(x_k,y_k)>e$. By Lemma 3.3, this implies that $f|_C$ is not chain continuous and thus $C\in {\mathcal {C}}_{no}(f|_{C(x)})$.

• (2) Since (x, y) is a scrambled pair for f, we have

  $$\begin{aligned} \liminf _{i\rightarrow \infty }d(f^i(x),f^i(y))=0; \end{aligned}$$

  therefore, there are a sequence $0\le i_1<i_2<\cdots$ and $z\in X$ such that

  $$\begin{aligned} \lim _{j\rightarrow \infty }d(f^{i_j}(x),f^{i_j}(y))=0 \end{aligned}$$

  and

  $$\begin{aligned} \lim _{j\rightarrow \infty }f^{i_j}(x)=z, \end{aligned}$$

  which implies $z\in \omega (x,f)\cap \omega (y,f)$ and so $\omega (y,f)\subset C$. By

  $$\begin{aligned} \omega (x,f)\cup \omega (y,f)\subset C, \end{aligned}$$

  we obtain

  $$\begin{aligned} \lim _{i\rightarrow \infty }d(f^i(x),C)=\lim _{i\rightarrow \infty }d(f^i(y),C)=0. \end{aligned}$$

  On the other hand, since (x, y) is a scrambled pair for f, we have

  $$\begin{aligned} \limsup _{i\rightarrow \infty }d(f^i(x),f^i(y))>0. \end{aligned}$$

  These condition clearly imply that $f|_C$ is not chain continuous and thus $C\in {\mathcal {C}}_{no}(f|_{C(x)})$.

• (3) Since $\omega (x,f)$ is not a minimal set for f, we have a closed f-invariant subset S of $\omega (x,f)$ such that

  $$\begin{aligned} \emptyset \ne S\ne \omega (x,f). \end{aligned}$$

  Fix $p\in S$, $q\in \omega (x,f)$, and $e>0$ with $d(p,q)>e$. Since $\omega (x,f)\subset C$ and $f|_C:C\rightarrow C$ is chain transitive, for any $\delta >0$, there are $\delta$-chains $(x_i)_{i=0}^k$ and $(y_i)_{i=0}^k$ of $f|_C$ with $x_0=y_0=y_k=p$ and $x_k=q$. This implies that $f|_C$ is not chain continuous and thus $C\in {\mathcal {C}}_{no}(f|_{C(x)})$.$\square$

Finally, we give an example mentioned in Remark 1.5. For a continuous map $f:X\rightarrow X$, $C\in {\mathcal {C}}(f)$ is said to be terminal if C is chain stable. The proof of the following lemma is left to the reader.

Lemma 3.4

Let $f:X\rightarrow X$ be a continuous map. For any $x\in X$ and $C\in {\mathcal {C}}(f)$ with $\omega (x,f)\subset C$, if C is terminal, then

$$\begin{aligned} C(x)=\{f^i(x):i\ge 0\}\cup C. \end{aligned}$$

Example 3.1

This example is taken from Kawaguchi (2021b). Let $\sigma :[-1,1]^{\mathbb {N}}\rightarrow [-1,1]^{\mathbb {N}}$ be the shift map and let d be the metric on $[-1,1]^{\mathbb {N}}$ defined by

$$\begin{aligned} d(x,y)=\sup _{i\ge 1}2^{-i}|x_i-y_i| \end{aligned}$$

for all $x=(x_i)_{i\ge 1},y=(y_i)_{i\ge 1}\in [-1,1]^{\mathbb {N}}$. Let $s=(s_k)_{k\ge 1}$ be a sequence of numbers with $1>s_1>s_2>\cdots$ and $\lim _{k\rightarrow \infty }s_k=0$. Put

$$\begin{aligned} S=\{0\}\cup \{-s_k:k\ge 1\}\cup \{s_k:k\ge 1\}, \end{aligned}$$

a closed subset of $[-1,1]$. We define a closed $\sigma$-invariant subset X of $S^{\mathbb {N}}$ by

$$\begin{aligned} X=\{x=(x_i)_{i\ge 1}\in S^{\mathbb {N}}:|x_1|\ge |x_2|\ge \cdots \}. \end{aligned}$$

Let $f=\sigma |_X:X\rightarrow X$, $X_k=\{-s_k,s_k\}^{\mathbb {N}}$ for each $k\ge 1$, and let $X_0=\{0^\infty \}$. Then, we have

$$\begin{aligned} CR(f)=\{x=(x_i)_{i\ge 1}\in X:|x_1|=|x_2|=\cdots \}=X_0\cup \bigcup _{k\ge 1}X_k \end{aligned}$$

and

$$\begin{aligned} {\mathcal {C}}(f)=\{X_0\}\cup \{X_k:k\ge 1\}. \end{aligned}$$

Note that $X_0$ is terminal. For a rapidly decreasing sequence $s=(s_k)_{k\ge 1}$, we can show that f satisfies the shadowing property and so $X=Sh(f)$. Let $x=(s_1,s_2,s_3,\dots )$ and note that $x\in X$. Since $\omega (x,f)=X_0$ and $X_0$ is terminal, by Lemma 3.4, we obtain

$$\begin{aligned} C(x)=\{f^i(x):i\ge 0\}\cup X_0. \end{aligned}$$

By Theorem 1.1, we see that

$$\begin{aligned} x\not \in \bigcup _{r>0}Ent_r(f). \end{aligned}$$

We shall show that $x\in Ent(f)$. Let

$$\begin{aligned} x_k=(s_1,s_2,\dots ,s_k,s_k,s_k,\dots ), \end{aligned}$$

$k\ge 1$, and note that $x_k\in X$ for each $k\ge 1$. For any $k\ge 1$, since $h(f,X_k)\ge \log {2}>0$, we have $h(f,X_k,r_k)>0$ and so $X_k\cap Ent_{r_k}(f)\ne \emptyset$ for some $r_k>0$. For every $k\ge 1$, since $X_k\subset C(x_k)$, we obtain $C(x_k)\cap Ent_{r_k}(f)\ne \emptyset$; therefore, Lemma 3.1 implies that $x_k\in Ent_{s_k}(f)$ for all $0<s_k<r_k$. In particular, we have $x_k\in Ent(f)$ for all $k\ge 1$. Since $\lim _{k\rightarrow \infty }x_k=x$, we conclude that $x\in Ent(f)$.

4 Proof of Theorem 1.4 and an example

In this section, we prove Theorem 1.4 and give an example mentioned in Sect. 1.

Proof of Theorem 1.4

First, we prove the implication $(1)\implies (2)$. If f is not h-expansive, then

$$\begin{aligned} h_f^*(\epsilon )=\sup _{x\in X}h(f,\Phi _{\epsilon }(x))>0 \end{aligned}$$

for all $\epsilon >0$. Given any $\epsilon >0$, there exists $x\in X$ such that $h(f,\Phi _{\epsilon /2}(x))>0$ and so we have $h(f,\Phi _{\epsilon /2}(x),r)>0$ for some $r>0$. For any $0<\Delta \le r$, we take an open cover ${\mathcal {U}}=\{U_i:1\le i\le m\}$ of X such that $d(a,b)\le \Delta$ for all $1\le i\le m$ and $a,b\in U_i$. Since

$$\begin{aligned} h(f,\Phi _{\epsilon /2}(x),r)=\limsup _{n\rightarrow \infty }\frac{1}{n}\log {s_n(f,\Phi _{\epsilon /2}(x),r)}>0, \end{aligned}$$

we have

$$\begin{aligned} s_n(f,\Phi _{\epsilon /2}(x),r)>m^2 \end{aligned}$$

for some $n\ge 1$. Then, there are $u,v\in \Phi _{\epsilon /2}(x)$ such that $d_n(u,v)>r$, $u,v\in U_{i_0}$, and $f^{n-1}(u),f^{n-1}(v)\in U_{i_{n-1}}$ for some $U_{i_0},U_{i_{n-1}}\in {\mathcal {U}}$. It follows that

$$\begin{aligned} \max \{d(u,v),d(f^{n-1}(u),f^{n-1}(v))\}\le \Delta \le r \end{aligned}$$

and

$$\begin{aligned} r<\max _{0\le i\le n-1}d(f^i(u),f^i(v))\le \epsilon . \end{aligned}$$

Since $0<\Delta \le r$ is arbitrary, this implies the existence of $r>0$ as in (2). Since $\epsilon >0$ is arbitrary, $(1)\implies (2)$ has been proved.

Next, we prove the implication $(2)\implies (3)$. The proof is similar to the proof of Lemma 3.1 in Artigue et al. (2020). Given any $\epsilon >0$, we choose $r>0$ as in (2). We fix $0<\gamma <\min \{\epsilon ,r/2\}$. Since f has the shadowing property, there is $\delta >0$ such that every $\delta$-pseudo orbit of f is $\gamma$-shadowed by some point of X. By the choice of r, we obtain a pair

$$\begin{aligned} (\alpha (0),\alpha (1))=((x_i)_{i=0}^k,(y_i)_{i=0}^k) \end{aligned}$$

of $\delta$-chains of f with $(x_0,x_k)=(y_0,y_k)$ and

$$\begin{aligned} r\le \max _{0\le i\le k}d(x_i,y_i)\le \epsilon . \end{aligned}$$

Then, the chain transitivity of f gives a $\delta$-chain $\beta =(z_i)_{i=0}^l$ of f with $z_0=x_k=y_k$ and $z_l=x_0=y_0$. For any $s=(s_n)_{n\ge 1}\in \{0,1\}^{\mathbb {N}}$, we consider a $\delta$-pseudo orbit

$$\begin{aligned} \Gamma (s)=\alpha (s_1)\beta \alpha (s_2)\beta \alpha (s_3)\beta \cdots \end{aligned}$$

of f. Let $m=k+l$,

$$\begin{aligned} Y=\{y\in X:\Gamma (s)\text { is }\gamma \text {-shadowed by }y\text { for some }s\in \{0,1\}^{\mathbb {N}}\}, \end{aligned}$$

and define a map $\pi :Y\rightarrow \{0,1\}^{\mathbb {N}}$ so that $\Gamma (\pi (y))$ is $\gamma$-shadowed by y for all $y\in Y$. By a standard argument, we can show that the following conditions are satisfied

• Y is a closed subset of X,

• $f^{m}(Y)\subset Y$,

• $\pi$ is well-defined,

• $\pi$ is surjective,

• $\pi$ is continuous,

• $\pi \circ f^m=\sigma \circ \pi$, where $\sigma :\{0,1\}^{\mathbb {N}}\rightarrow \{0,1\}^{\mathbb {N}}$ is the shift map.

It follows that Y is a closed $f^m$-invariant subset of Y and

$$\begin{aligned} \pi :(Y,f^m)\rightarrow (\{0,1\}^{\mathbb {N}},\sigma ) \end{aligned}$$

is a factor map. By the definition of $\Gamma (s)$, $s\in \{0,1\}^{\mathbb {N}}$, we see that

$$\begin{aligned} \sup _{i\ge 0}d(f^i(x),f^i(y))\le 3\epsilon \end{aligned}$$

for all $x,y\in Y$. Since $\epsilon >0$ is arbitrary, $(2)\implies (3)$ has been proved.

We shall prove the implication $(3)\implies (1)$. Given any $\epsilon >0$, we take $m\ge 1$ and Y as in (3). Take $p\in Y$ and note that $Y\subset \Phi _\epsilon (p)$. It follows that

$$\begin{aligned} h_f^*(\epsilon )\ge h(f,\Phi _\epsilon (p))\ge h(f,Y)\ge \frac{1}{m}h(f^m,Y)\ge \frac{1}{m}h(\sigma ,\{0,1\}^{\mathbb {N}})=\frac{1}{m}\log {2}>0. \end{aligned}$$

Since $\epsilon >0$ is arbitrary, $(3)\implies (1)$ has been proved.

The implication $(4)\implies (2)$ is obvious from the definitions. It remains to prove the implication $(3)\implies (4)$. Given any $\epsilon >0$, we take $m\ge 1$ and Y as in (3). Since

$$\begin{aligned} h(f^m,Y)\ge h(\sigma ,\{0,1\}^{\mathbb {N}})=\log {2}>0, \end{aligned}$$

by Corollary 2.4 of Blanchard et al. (2002), there is a scrambled pair $(x,y)\in Y^2$ for $f^m$. Then, (x, y) is also a scrambled pair for f and satisfies

$$\begin{aligned} \sup _{i\ge 0}d(f^i(x),f^i(y))\le \epsilon \end{aligned}$$

because $x,y\in Y$. Since $\epsilon >0$ is arbitrary, $(4)\implies (3)$ has been proved. This completes the proof of Theorem 1.4. $\square$

We use Theorem 1.4 to obtain a counter-example for a question in Artigue et al. (2022). The example will be given as an inverse limit of the full-shift $(\{0,1\}^{\mathbb {Z}},\sigma )$ with respect to a factor map

$$\begin{aligned} F:(\{0,1\}^{\mathbb {Z}},\sigma )\rightarrow (\{0,1\}^{\mathbb {Z}},\sigma ). \end{aligned}$$

We need three auxiliary lemmas. A homeomorphism $f:X\rightarrow X$ is said to be expansive if there is $e>0$ such that

$$\begin{aligned} \Gamma _e(x)=\{x\} \end{aligned}$$

for all $x\in X$ and such e is called an expansive constant for f. It is known that for a homeomorphism $f:X\rightarrow X$ with an expansive constant $e>0$ and $x,y\in X$, if

$$\begin{aligned} \sup _{i\ge 0}d(f^i(x),f^i(y))\le e, \end{aligned}$$

then (x, y) is an asymptotic pair for f. The following lemma gives a sufficient condition for an inverse limit system to be h-expansive.

Lemma 4.1

Let $\pi =(\pi _n:(X_{n+1},f_{n+1})\rightarrow (X_n,f_n))_{n\ge 1}$ be a sequence of factor maps such that for every $n\ge 1$, $f_n:X_n\rightarrow X_n$ is an expansive transitive homeomorphism with the shadowing property. Let

$$\begin{aligned} (Y,g)=\lim _\pi (X_n,f_n) \end{aligned}$$

and note that g is a transitive homeomorphism with the shadowing property. If g is not h-expansive, then for any $N\ge 1$, there are $M\ge N$ and a scrambled pair $(x_{M+1},y_{M+1})\in X_{M+1}^2$ for $f_{M+1}$ such that $(\pi _M(x_{M+1}),\pi _M(y_{M+1}))$ is an asymptotic pair for $f_M$.

Proof

Let D be a metric on Y. Let $d_n$ be a metric on $X_n$ and $e_n>0$ be an expansive constant for $f_n$ for each $n\ge 1$. Given any $N\ge 1$, we take $\epsilon _N>0$ such that for any $p=(p_n)_{n\ge 1}$, $q=(q_n)_{n\ge 1}\in Y$, $D(p,q)\le \epsilon _N$ implies $d_n(p_n,q_n)\le e_n$ for all $1\le n\le N$. Since g is not h-expansive, by Theorem 1.4, there is a scrambled pair

$$\begin{aligned} (x,y)=((x_n)_{n\ge 1},(y_n)_{n\ge 1})\in Y^2 \end{aligned}$$

for g with

$$\begin{aligned} \sup _{i\ge 0}D(g^i(x),g^i(y))\le \epsilon _N. \end{aligned}$$

Then, for every $1\le n\le N$, since

$$\begin{aligned} \sup _{i\ge 0}d_n(f_n^i(x_n),f_n^i(y_n))=\sup _{i\ge 0}d_n(g^i(x)_n,g^i(y)_n)\le e_n, \end{aligned}$$

$(x_n,y_n)$ is an asymptotic pair for $f_n$. Since (x, y) is a scrambled pair for g and so a proximal for g, $(x_n,y_n)$ is a proximal pair for $f_n$ for all $n\ge 1$. If $(x_n,y_n)$ is an asymptotic pair for $f_n$ for all $n\ge 1$, then (x, y) is an asymptotic pair for g, which is a contradiction. Thus, there is $m\ge 1$ such that $(x_{m+1},y_{m+1})$ is a scrambled pair for $f_{m+1}$. Letting

$$\begin{aligned} M=\min \{m\ge 1:(x_{m+1},y_{m+1})\text { is a scrambled pair for }f_{m+1}\}, \end{aligned}$$

we see that $M\ge N$, $(x_{M+1},y_{M+1})$ is a scrambled pair for $f_{M+1}$, and $(x_M,y_M)$ is an asymptotic pair for $f_M$, thus the lemma has been proved. $\square$

A map $F:X\rightarrow X$ is said to be an open map if for any open subset U of X, f(U) is an open subset of X. Any continuous open map $F:X\rightarrow X$ satisfies the following property: for every $r>0$, there is $\delta >0$ such that for any $s,t\in X$ with $d(s,t)\le \delta$ and $u\in F^{-1}(s)$, we have $d(u,v)\le r$ for some $v\in F^{-1}(t)$.

For a continuous map $f:X\rightarrow X$, a sequence $(x_i)_{i\ge 0}$ of points in X is called a limit-pseudo orbit of f if

$$\begin{aligned} \lim _{i\rightarrow \infty }d(f(x_i),x_{i+1})=0, \end{aligned}$$

and said to be limit shadowed by $x\in X$ if

$$\begin{aligned} \lim _{i\rightarrow \infty }d(f^i(x),x_i)=0. \end{aligned}$$

The next lemma is needed for the proof of Lemma 4.3.

Lemma 4.2

Let $f:X\rightarrow X$ be a homeomorphism and let $F:(X,f)\rightarrow (X,f)$ be a factor map such that

• (1) F is an open map,

• (2) $d(v,v')\ge 1$ for all $t\in X$ and $v,v'\in F^{-1}(t)$ with $v\ne v'$.

Suppose that

• (3) $(x_i)_{i\ge 0}$ is a limit-pseudo orbit of f and limit-shadowed by $x\in X$,

• (4) $(z_i)_{i\ge 0}$ is a limit-pseudo orbit of f with $z_i\in F^{-1}(x_i)$ for all $i\ge 0$.

Then, there is $z\in F^{-1}(x)$ such that $(z_i)_{i\ge 0}$ is limit-shadowed by z.

Proof

By (3), letting $\delta _i=d(x_i,f^i(x))$, $i\ge 0$, we have $\lim _{i\rightarrow \infty }\delta _i=0$. By (1), we can take a sequence $r_i>0$, $i\ge 0$, such that

• $\lim _{i\rightarrow \infty }r_i=0$,

• for any $i\ge 0$, $s,t\in X$ with $d(s,t)\le \delta _i$, and $u\in F^{-1}(s)$, we have $d(u,v)\le r_i$ for some $v\in F^{-1}(t)$.

With use of (4), we fix $N\ge 0$ satisfying the following conditions

• $0<r_i<1/2$ for all $i\ge N$,

• $d(u,v)\le r_i$ implies $d(f(u),f(v))\le 1/4$ for all $i\ge N$ and $u,v\in X$,

• $d(f(z_i),z_{i+1})\le 1/4$ for all $i\ge N$.

By $\delta _N=d(x_N,f^N(x))$ and $z_N\in F^{-1}(x_N)$, we obtain $w_N\in F^{-1}(f^N(x))$ with $d(z_N,w_N)\le r_N$. Note that

$$\begin{aligned} F(f^j(w_N))=f^j(F(w_N))=f^j(f^N(x))=f^{N+j}(x) \end{aligned}$$

for every $j\ge 0$. By induction on j, we prove that $d(z_{N+j},f^j(w_N))\le r_{N+j}$ for all $j\ge 0$. Assume that $d(z_{N+j},f^j(w_N))\le r_{N+j}$ for some $j\ge 0$. Then,

$$\begin{aligned} d(z_{N+j+1},f^{j+1}(w_N))&\le d(z_{N+j+1},f(z_{N+j}))+d(f(z_{N+j}),f(f^j(w_N)))\\&\le 1/4+1/4=1/2. \end{aligned}$$

Since

$$\begin{aligned} \delta _{N+j+1}=d(x_{N+j+1},f^{N+j+1}(x)) \end{aligned}$$

and $z_{N+j+1}\in F^{-1}(x_{N+j+1})$, we have

$$\begin{aligned} d(z_{N+j+1},w)\le r_{N+j+1} \end{aligned}$$

for some $w\in F^{-1}(f^{N+j+1}(x))$. Since $f^{j+1}(w_N)\in F^{-1}(f^{N+j+1}(x))$, by (2), we obtain

$$\begin{aligned} d(z_{N+j+1},w')&\ge d(f^{j+1}(w_N),w')-d(z_{N+j+1},f^{j+1}(w_N))\\&\ge 1-1/2=1/2>r_{N+j+1} \end{aligned}$$

for all $w'\in F^{-1}(f^{N+j+1}(x))$ with $w'\ne f^{j+1}(w_N)$. It follows that $w=f^{j+1}(w_N)$ and so

$$\begin{aligned} d(z_{N+j+1},f^{j+1}(w_N))\le r_{N+j+1}; \end{aligned}$$

therefore, the induction is complete. Let $z=f^{-N}(w_N)$ and note that

$$\begin{aligned} f^N(F(z))=F(f^N(z))=F(w_N)=f^N(x). \end{aligned}$$

Since f is a homeomorphism, we have $F(z)=x$, that is, $z\in F^{-1}(x)$. Moreover, we obtain

$$\begin{aligned}{} & {} \lim _{i\rightarrow \infty }d(z_i,f^i(z)) \\{} & {} \quad =\lim _{j\rightarrow \infty }d(z_{N+j},f^{N+j}(z))=\lim _{j\rightarrow \infty }d(z_{N+j},f^j(w_N))=\lim _{j\rightarrow \infty }r_{N+j}=0, \end{aligned}$$

thus the lemma has been proved. $\square$

The following lemma gives a sufficient condition for an inverse limit system to satisfy the s-limit shadowing property.

Lemma 4.3

Let $f:X\rightarrow X$ be a homeomorphism and let $F:(X,f)\rightarrow (X,f)$ be a factor map such that

• (1) F is an open map,

• (2) $d(v,v')\ge 1$ for all $t\in X$ and $v,v'\in F^{-1}(t)$ with $v\ne v'$.

Let $(X_n,f_n)=(X,f)$ and $\pi _n=F:(X,f)\rightarrow (X,f)$ for all $n\ge 1$. Let

$$\begin{aligned} (Y,g)=\lim _\pi (X_n,f_n). \end{aligned}$$

If f has the s-limit shadowing property, then g satisfies the s-limit shadowing property.

Proof

Let D be a metric on Y. Given any $\epsilon >0$, we take $N\ge 1$ and $\epsilon _N>0$ such that for any $p=(p_n)_{n\ge 1}$, $q=(q_n)_{n\ge 1}\in Y$, $d(p_N,q_N)\le \epsilon _N$ implies $D(p,q)\le \epsilon$. Since f has the s-limit shadowing property, there is $\delta _N>0$ such that every $\delta _N$-limit-pseudo orbit of f is $\epsilon _N$-limit shadowed by some point of X. We take $\delta >0$ such that $D(p,q)\le \delta$ implies $d(p_N,q_N)\le \delta _N$ for all $p=(p_n)_{n\ge 1}$, $q=(q_n)_{n\ge 1}\in Y$. Let $\xi =(x^{(i)})_{i\ge 0}$ be a $\delta$-limit-pseudo orbit of g. Then, for every $i\ge 0$, since $D(g(x^{(i)}),x^{(i+1)})\le \delta$, we have

$$\begin{aligned} d(f(x_N^{(i)}),x_N^{(i+1)})=d(g(x^{(i)})_N,x_N^{(i+1)})\le \delta _N. \end{aligned}$$

Also, since $\lim _{i\rightarrow \infty }D(g(x^{(i)}),x^{(i+1)})=0$, we have

$$\begin{aligned} \lim _{i\rightarrow \infty }d(f(x_n^{(i)}),x_n^{(i+1)})=\lim _{i\rightarrow \infty }d(g(x^{(i)})_n,x_n^{(i+1)})=0 \end{aligned}$$

for all $n\ge 1$. It follows that $(x_N^{(i)})_{i\ge 0}$ is a $\delta _N$-limit-pseudo of f and so $\epsilon _N$-limit shadowed by some $x_N\in X$. Then, since

$$\begin{aligned} \lim _{i\rightarrow \infty }d(f(x_N^{(i)}),x_N^{(i+1)})=\lim _{i\rightarrow \infty }d(f^i(x_N),x_N^{(i)})=0 \end{aligned}$$

and

$$\begin{aligned} \lim _{i\rightarrow \infty }d(f(x_{N+1}^{(i)}),x_{N+1}^{(i+1)})=0, \end{aligned}$$

by Lemma 4.2, we have

$$\begin{aligned} \lim _{i\rightarrow \infty }d(f^i(x_{N+1}),x_{N+1}^{(i)})=0 \end{aligned}$$

for some $x_{N+1}\in F^{-1}(x_N)$. Inductively, we obtain $x_{N+k}\in X$, $k\ge 0$, such that

$$\begin{aligned} \lim _{i\rightarrow \infty }d(f^i(x_{N+k}),x_{N+k}^{(i)})=0 \end{aligned}$$

and $x_{N+k+1}\in F^{-1}(x_{N+k})$ for all $k\ge 0$. We define $y=(y_n)_{n\ge 1}\in Y$ by

$$\begin{aligned} y_n= {\left\{ \begin{array}{ll} F^{N-n}(x_N)&{}\text {for all }1\le n\le N\\ x_n&{}\text {for all }n\ge N \end{array}\right. }. \end{aligned}$$

Given any $i\ge 0$, by

$$\begin{aligned} d(g^i(y)_N,x_N^{(i)})=d(f^i(y_N),x_N^{(i)})=d(f^i(x_N),x_N^{(i)})\le \epsilon _N, \end{aligned}$$

we obtain

$$\begin{aligned} D(g^i(y),x^{(i)})\le \epsilon . \end{aligned}$$

Moreover, since

$$\begin{aligned} \lim _{i\rightarrow \infty }d(g^i(y)_n,x_n^{(i)})&=\lim _{i\rightarrow \infty }d(f^i(y_n),x_n^{(i)})\\&=\lim _{i\rightarrow \infty }d(f^i(F^{N-n}(x_N)),x_n^{(i)})\\&=\lim _{i\rightarrow \infty }d(F^{N-n}(f^i(x_N)),F^{N-n}(x_N^{(i)}))=0 \end{aligned}$$

for all $1\le n\le N$; and

$$\begin{aligned} \lim _{i\rightarrow \infty }d(g^i(y)_{N+k},x_{N+k}^{(i)})&=\lim _{i\rightarrow \infty }d(f^i(y_{N+k}),x_{N+k}^{(i)})\\&=\lim _{i\rightarrow \infty }d(f^i(x_{N+k}),x_{N+k}^{(i)})=0 \end{aligned}$$

for all $k\ge 0$, we obtain

$$\begin{aligned} \lim _{i\rightarrow \infty }D(g^i(y),x^{(i)})=0. \end{aligned}$$

In other words, $\xi$ is $\epsilon$-limit shadowed by y. Since $\epsilon >0$ is arbitrary, we conclude that g satisfies the s-limit shadowing property, completing the proof of the lemma. $\square$

Finally, we give the example.

Example 4.1

Let ${\mathbb {Z}}_2=\{0,1\}$. We define a metric d on $\{0,1\}^{\mathbb {Z}}$ by

$$\begin{aligned} d(x,y)=\sup _{n\in {\mathbb {Z}}}2^{-|n|}|x_n-y_n| \end{aligned}$$

for all $x=(x_n)_{n\in {\mathbb {Z}}}$, $y=(y_n)_{n\in {\mathbb {Z}}}\in \{0,1\}^{\mathbb {Z}}$. Note that the shift map

$$\begin{aligned} \sigma :\{0,1\}^{\mathbb {Z}}\rightarrow \{0,1\}^{\mathbb {Z}} \end{aligned}$$

is an expansive mixing homeomorphism with the shadowing property and so satisfies the s-limit shadowing property (see, e.g.  Barwell et al. 2012). We define a map $F:\{0,1\}^{\mathbb {Z}}\rightarrow \{0,1\}^{\mathbb {Z}}$ by for any $x=(x_n)_{n\in {\mathbb {Z}}}$, $y=(y_n)_{n\in {\mathbb {Z}}}\in \{0,1\}^{\mathbb {Z}}$, $y=F(x)$ if and only if

$$\begin{aligned} y_n=x_n+x_{n+1} \end{aligned}$$

for all $n\in {\mathbb {Z}}$. Note that F gives a factor map

$$\begin{aligned} F:(\{0,1\}^{\mathbb {Z}},\sigma )\rightarrow (\{0,1\}^{\mathbb {Z}},\sigma ). \end{aligned}$$

Given any $x=(x_n)_{n\in {\mathbb {Z}}},y=(y_n)_{n\in {\mathbb {Z}}},z=(z_n)_{n\in {\mathbb {Z}}},w=(w_n)_{n\in {\mathbb {Z}}}\in \{0,1\}^{\mathbb {Z}}$, assume that

• (x, y) is an asymptotic pair for $\sigma$,

• $(F(z),F(w))=(x,y)$.

Then, there is $N\ge 0$ such that $x_n=y_n$ for all $n\ge N$. If $z_N=w_N$, we have $z_n=w_n$ for all $n\ge N$ and so (z, w) is an asymptotic pair for $\sigma$. If $z_N\ne w_N$, we have $z_n\ne w_n$ for all $n\ge N$ and so

$$\begin{aligned} \liminf _{i\rightarrow \infty }d(\sigma ^i(z),\sigma ^i(w))=1>0, \end{aligned}$$

thus (z, w) is a distal pair for $\sigma$. In both cases, (z, w) is not a scrambled pair for $\sigma$.

For any $m\ge 1$ and $a=(a_n)_{n=-m}^m\in \{0,1\}^{2\,m+1}$, we define $b=(b_n)_{n=-m}^{m-1}\in \{0,1\}^{2m}$ by

$$\begin{aligned} b_n=a_n+a_{n+1} \end{aligned}$$

for all $-m\le n\le m-1$. Letting

$$\begin{aligned} S(a)=\{x=(x_n)_{n\in {\mathbb {Z}}}:x_n=a_n\,\text { for all }-m\le n\le m\} \end{aligned}$$

and

$$\begin{aligned} T(b)=\{x=(x_n)_{n\in {\mathbb {Z}}}:x_n=b_n\,\text { for all }-m\le n\le m-1\}, \end{aligned}$$

we obtain $F(S(a))=T(b)$, an open subset of $\{0,1\}^{\mathbb {Z}}$. Since $m\ge 1$ and $a=(a_n)_{n=-m}^m\in \{0,1\}^{2\,m+1}$ are arbitrary, it follows that F is an open map. Given any $y=(y_n)_{n\in {\mathbb {Z}}}\in \{0,1\}^{\mathbb {Z}}$, we define ${\hat{y}}=({\hat{y}}_n)_{n\in {\mathbb {Z}}}\in \{0,1\}^{\mathbb {Z}}$ by ${\hat{y}}_n=y_n+1$ for all $n\in {\mathbb {Z}}$. Then, for any $x\in \{0,1\}^{\mathbb {Z}}$, taking $y\in F^{-1}(x)$, we have $F^{-1}(x)=\{y,{\hat{y}}\}$. Note that $d(y,{\hat{y}})=1$ for all $y\in \{0,1\}^{\mathbb {Z}}$.

Let $(X_n,f_n)=(\{0,1\}^{\mathbb {Z}},\sigma )$ and $\pi _n=F:(\{0,1\}^{\mathbb {Z}},\sigma )\rightarrow (\{0,1\}^{\mathbb {Z}},\sigma )$ for all $n\ge 1$. Let

$$\begin{aligned} (Y,g)=\lim _\pi (X_n,f_n) \end{aligned}$$

and let D be a metric on Y. Since $\{0,1\}^{\mathbb {Z}}$ is totally disconnected and $\sigma :\{0,1\}^{\mathbb {Z}}\rightarrow \{0,1\}^{\mathbb {Z}}$ is a mixing homeomorphism,

• Y is totally disconnected,

• g is a mixing homeomorphism.

By Lemmas 4.1 and 4.3, we obtain the following properties

• g is h-expansive,

• g has the s-limit shadowing property.

We shall show that

• g is not countably-expansive,

• g satisfies $Y_e=\emptyset$, where

  $$\begin{aligned} Y_e=\{q\in Y:\Gamma _\epsilon (q)=\{q\}\,\text { for some }\epsilon >0\}. \end{aligned}$$

Let $q=(q_n)_{n\ge 1}\in Y$ and $N\ge 1$. Let $F^{-1}(x)=\{x^a,x^b\}$ for all $x\in \{0,1\}^{\mathbb {Z}}$. Then, for all $c=(c_k)_{k\ge 1}\in \{a,b\}^{\mathbb {N}}$, we define $q(c)=(q(c)_n)_{n\ge 1}\in Y$ by $q(c)_n=q_n$ for all $1\le n\le N$; and

$$\begin{aligned} q(c)_{N+k}=q(c)_{N+k-1}^{c_k} \end{aligned}$$

for all $k\ge 1$. Given any $\epsilon >0$, if $N\ge 1$ is large enough, q(c), $c\in \{a,b\}^{\mathbb {N}}$, satisfies $q(c)\in \Gamma _\epsilon (q)$ for all $c\in \{a,b\}^{\mathbb {N}}$. Since

$$\begin{aligned} \{q(c):c\in \{a,b\}^{\mathbb {N}}\} \end{aligned}$$

is an uncountable set, it follows that g is not countably-expansive. Since $q\in Y$ and $\epsilon >0$ are arbitrary, it also follows that $Y_e=\emptyset$.