Distributionally chaotic maps are $C^0$-dense

TL;DR

This paper proves that continuous maps exhibiting distributional chaos of type 1 are densely present in the space of all continuous self-maps on any compact topological manifold, highlighting the prevalence of chaotic behavior.

Contribution

It establishes the $C^0$-density of DC1 maps in the space of continuous self-maps on compact manifolds, extending understanding of chaos distribution.

Findings

DC1 maps are $C^0$-dense in continuous self-maps

Chaotic behavior is prevalent in compact manifolds

Extends previous results on distributional chaos

Abstract

We prove that the set of maps which exhibit distributional chaos of type 1 (DC1) is $C^0$-dense in the space of continuous self-maps of given any compact topological manifold (possibly with boundary).

Equations

$$d_{C^0}(f,g)=\underset{x\in X}{\sup }d(f(x),g(x))$$

$$\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n}|\{0\le i\le n-1:d(f^i(x),f^i(y))<\delta \}|=1\text{for every δ>0},$$

$$\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n}|\{0\le i\le n-1:d(f^i(x),f^i(y))>{\delta }_0\}|=1\text{for some δ0>0}.$$

$$E_K(M)=\{f\in C(M):h_{top}(f)\ge K\}$$

$$\mathcal{D}(\delta )=\{D_{i,j}:1\le i\le K,0\le j\le m_i-1\}$$

$$R=\{(a,a)\in X^2:a\in X\}\cup \{(a,b)\in CR(f{)}^2:\text{{a,b}⊂A for some A∈D(δ)}\}.$$

$$\frac{a_n(m_n-2)+1}{b_n+a_n{m}_n+1}>1-n^{-1}$$

$$|\{0\le i\le c_n-1:d(f^i(x_j),x_{j,i})<\frac{\delta }{2}\}|\ge c_n(1-2{\delta }^{-1}{ϵ}_{j,c_n})$$

$$\{b_n\le i\le b_n+a_n{m}_n\}\subset \{0\le i\le c_n-1:x_{0,i}=x_{1,i}\},$$

$$|\{0\le i\le c_n-1:x_{0,i}=x_{1,i}\}|\ge c_n\cdot \frac{a_n{m}_n+1}{c_n}>c_n(1-n^{-1}).$$

$$|\{0\le i\le c_n-1:d(f^i(x_0),f^i(x_1))<\delta \}|>c_n(1-n^{-1}-2{\delta }^{-1}{ϵ}_{0,c_n}-2{\delta }^{-1}{ϵ}_{1,c_n}).$$

$$\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n}|\{0\le i\le n-1:d(f^i(x_0),f^i(x_1))<\delta \}|=1.$$

$$\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n}|\{0\le i\le n-1:d(f^i(x_0),f^i(x_1))>\frac{r}{3}\}|=1.$$

$$|\{0\le i\le c_n-1:d(f^i(x_j),x_{j,i})<\frac{r}{3}\}|\ge c_n(1-3r^{-1}{ϵ}_{j,c_n})$$

$$\{b_n+a_n\le i\le b_n+a_n+a_n(m_n-2)\}\subset \{0\le i\le c_n-1:d(x_{0,i},x_{1,i})>r\},$$

$$|\{0\le i\le c_n-1:d(x_{0,i},x_{1,i})>r\}|\ge c_n\cdot \frac{a_n(m_n-2)+1}{c_n}>c_n(1-n^{-1}).$$

$$|\{0\le i\le c_n-1:d(f^i(x_0),f^i(x_1))>\frac{r}{3}\}|>c_n(1-n^{-1}-3r^{-1}{ϵ}_{0,c_n}-3r^{-1}{ϵ}_{1,c_n}).$$

$$Y=\{x\in CR(f):\text{ξ(s) is ϵ-shadowed by x for some s∈{0,1}N}\}$$

$$h(f,\{B_{ϵ}(x{)}^c,B_{ϵ}(y{)}^c\})\ge a^{-1}\log 2>0.$$

$$E(X,f)=\{(x,y)\in CR(f{)}^2:x\ne y\text{and}x\sim y\}.$$

$$E(Y,f^a)\subset E(X,f^a)\subset E(X,f),$$

$$(z,w)\in \omega (p,f^a)\times \omega (q,f^a)\subset {\pi }^{-1}(0^{\infty })\times {\pi }^{-1}(1^{\infty })\subset B_{ϵ}(x)\times B_{ϵ}(y).$$

$$\underset{j\to \infty }{\lim }(f^{n_j}(x(u)),f^{n_j}(x(v)))=(z,w).$$

$$\underset{n\to \infty }{\lim }d(f^n(x),f^n(A_x))=0\text{and}\underset{n\to \infty }{\lim }d(f^n(y),f^n(A_y))=0.$$

$$T=\{n\ge 0:d(f^n(x),f^n(y))>{\delta }_0\}$$

$$\underset{n\to \infty }{\mathrm{liminf}}d(f^n(x),f^n(y))=0\text{and}\underset{n\to \infty }{\mathrm{limsup}}d(f^n(x),f^n(y))>0.$$

Full text

1. Introduction

Throughout this paper, $X$ denotes a compact metric space endowed with a metric $d$. We denote by $C(X)$ the space of continuous self-maps of $X$, equipped with the metric $d_{C^{0}}$ defined by

[TABLE]

for $f,g\in C(X)$. The aim of this paper is to prove the following theorem.

Theorem 1.1**.**

Let $M$ be a compact topological manifold (possibly with boundary). Then, the set of maps which exhibit distributional chaos of type 1 (DC1) is dense in $C(M)$.

The notion of distributional chaos was introduced by Schweizer and Smítal [14]. For a continuous map $f:X\to X$, a pair of points $(x,y)\in X^{2}$ is said to be a DC1-pair for $f$ if

[TABLE]

and

[TABLE]

Then, $f$ is said to exhibit distributional chaos of type 1 (DC1) if there is an uncountable set $S\subset X$ such that for any $x,y\in S$ with $x\neq y$, $(x,y)$ is a DC1-pair for $f$. Indeed, the distributional chaos has two more versions: DC2 and DC3. They are numbered in the order of decreasing strength, so DC1 is the strongest, and DC2 is stronger than Li-Yorke chaos.

Hereafter, $M$ denotes a compact topological manifold (possibly with boundary). For any $f\in C(X)$, we denote by $h_{top}(f)$ the topological entropy of $f$. In [16], it was proved by Yano that a generic $f\in C(M)$ satisfies $h_{top}(f)=\infty$. Downarowicz proved that for any $f\in C(X)$, $h_{top}(f)>0$ implies DC2 [3]. These together imply that DC2 is generic in $C(M)$. Then, it is natural to ask whether DC1 is still generic in $C(M)$ or not. When $M$ is the unit interval (denoted by $I$), we know that for any $f\in C(I)$, $h_{top}(f)>0$ iff $f$ exhibits DC1 [14]; therefore, DC1 is generic in $C(I)$. Theorem 1.1 gives a partial answer to the question for a general $M$. It should be noticed that Pikuła showed that $h_{top}(f)>0$ does not necessarily imply DC1 for any $f\in C(X)$ [12]. This indicates that some additional assumptions besides positive topological entropy are needed to ensure DC1 for general continuous maps.

Shadowing is a natural candidate for such an assumption. Recently in [8], Li et al. proved that for any $f\in C(X)$ with the shadowing property, $f$ exhibits DC1 if one of the following properties holds: (1) $f$ is non-periodic transitive and has a periodic point, or (2) $f$ is non-trivial weakly mixing. Here, note that we have $h_{top}(f)>0$ in both cases. According to Mazur and Oprocha [10], we know that the shadowing property is generic in $C(M)$. However, because of the additional assumption (1) or (2), it is not obvious that these results imply the genericity of DC1.

The limit shadowing property introduced by Eirola et al. in [4] is a variant of the shadowing property defined as follows. Given a continuous map $f:X\to X$, a sequence $(x_{i})_{i\geq 0}$ of points in $X$ is a limit pseudo orbit of $f$ if $\lim_{i\to\infty}d(f(x_{i}),x_{i+1})=0$. Then, $f$ is said to have the limit shadowing property if for any limit pseudo orbit $(x_{i})_{i\geq 0}$ of $f$, there is $x\in X$ such that $\lim_{i\to\infty}d(f^{i}(x),x_{i})=0$, where such $x$ is called a limit shadowing point of $(x_{i})_{i\geq 0}$. The property provides a method of gaining information about the asymptotic behavior of the true orbits from pseudo-orbits of continuous maps. Then, we prove the following theorem.

Theorem 1.2**.**

Suppose that a continuous map $f:X\to X$ satisfies the following properties.

• (1)

The limit shadowing property.

• (2)

$h_{top}(f)>0$.

Then, $f$ exhibits distributional chaos of type 1 (DC1). Moreover, $E(X,f)\subset\overline{DC1(X,f)}$, where $E(X,f)$ (resp. $DC1(X,f)$) is the set of entropy pairs (resp. DC1-pairs) for $f$.

This theorem guarantees DC1 for any $f\in C(X)$ with the limit shadowing property without any other assumption than the positive topological entropy. In [10], it was also proved that the set of maps with the s-limit shadowing property is dense in $C(M)$. We know that the s-limit shadowing property always implies the limit shadowing property [1]. On the other hand, Yano proved that for any $K>0$, the set

[TABLE]

contains an open and dense subset of $C(M)$ [16]. Thus, Theorem 1.1 is implied by Theorem 1.2.

This paper consists of four sections. Some basic definitions and facts are briefly collected in Section 2. In Section 3, we introduce a relation on the chain recurrent set, and prove a few lemmas needed for the proof of Theorem 1.2. Then, Theorem 1.2 is proved in Section 4.

2. Preliminaries

In this section, we collect some basic definitions and facts used in this paper.

2.1. Chains, cycles, pseudo-orbits, and the shadowing property

Given a continuous map $f:X\to X$, a finite sequence $(x_{i})_{i=0}^{k}$ of points in $X$, where $k$ is a positive integer, is called a $\delta$-chain of $f$ if $d(f(x_{i}),x_{i+1})\leq\delta$ for every $0\leq i\leq k-1$. A $\delta$-chain $(x_{i})_{i=0}^{k}$ of $f$ is said to be a $\delta$-cycle of $f$ if $x_{0}=x_{k}$. For $\delta>0$, a sequence $(x_{i})_{i\geq 0}$ of points in $X$ is called a $\delta$-pseudo orbit of $f$ if $d(f(x_{i}),x_{i+1})\leq\delta$ for all $i\geq 0$. Then, for $\epsilon>0$, a $\delta$-pseudo orbit $(x_{i})_{i\geq 0}$ of $f$ is said to be $\epsilon$-shadowed by $x\in X$ if $d(f^{i}(x),x_{i})\leq\epsilon$ for all $i\geq 0$. 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 $X$. A point $x\in X$ is said to be a chain recurrent point for $f$ if for any $\delta>0$, there is a $\delta$-cycle $(x_{i})_{i=0}^{k}$ of $f$ such that $x_{0}=x_{k}=x$. The set of chain recurrent points for $f$ is denoted by $CR(f)$.

2.2. Entropy pairs

Given a continuous map $f:X\to X$ and an open cover $\mathcal{U}$ of $X$, we denote by $h(f,\mathcal{U})$ the entropy of $f$ relative to $\mathcal{U}$ (see [15] for details). The notion of entropy pairs was introduced by Blanchard [2]. A pair of points $(x,y)\in X^{2}$ with $x\neq y$ is said to be an entropy pair for $f$ if for any closed neighborhoods $A$ of $x$ and $B$ of $y$, we have $h(f,\{A^{c},B^{c}\})>0$ whenever $A\cap B=\emptyset$. The set of entropy pairs for $f$ is denoted by $E(X,f)\>(\subset X^{2})$.

For two continuous maps $f:X\to X$ and $g:Y\to Y$, we say that $(Y,g)$ is a factor of $(X,f)$ if there is a surjective continuous map $\pi:X\to Y$ such that $\pi\circ f=g\circ\pi$. Such a map is called a factor map, and also denoted as $\pi:(X,f)\to(Y,g)$. The basic properties of entropy pairs is summarized in the following lemma.

Lemma 2.1**.**

[2]** Given a factor map $\pi:(X,f)\to(Y,g)$, we have the following properties.

• (1)

$h_{top}(f)>0$* if and only if $E(X,f)\neq\emptyset$.*

• (2)

For any $x,y\in X$, if $(x,y)\in E(X,f)$ and $\pi(x)\neq\pi(y)$, then $(\pi(x),\pi(y))\in E(Y,g)$.

• (3)

For any $(z,w)\in E(Y,g)$, there is $(x,y)\in\pi^{-1}(z)\times\pi^{-1}(w)$ such that $(x,y)\in E(X,f)$.

We say that a continuous map $f:X\to X$ has upe if for any $(x,y)\in X^{2}$ with $x\neq y$, $(x,y)\in E(X,f)$. It is known that if $f$ has upe, then $f$ is weakly mixing, and when $f$ has the shadowing property, the converse holds [9]. By this, especially, the shift map $\sigma:\{0,1\}^{\mathbb{N}}\to\{0,1\}^{\mathbb{N}}$ has upe.

3. A chain relation and a few lemmas

In this section, we prove a few lemmas needed for the proof of Theorem 1.2. First, for any continuous map $f:X\to X$, we define a chain relation $\sim$ on $CR(f)$ as follows.

Definition 3.1**.**

For any $x,y\in CR(f)$, $x\sim y$ if and only if for every $\delta>0$, there are integers $m=m(\delta)>0$ and $N=N(\delta)>0$ such that for any integer $n\geq N$, there are two $\delta$-chains $(x_{i})_{i=0}^{mn}$, $(y_{i})_{i=0}^{mn}\subset CR(f)$ of $f$ such that $x_{0}=y_{mn}=x$ and $x_{mn}=y_{0}=y$.

Remark 3.1**.**

Even if we replace ‘$\subset CR(f)$’ with ‘$\subset X$’ in the above definition, the relation $\sim$ does not change. This can be seen as follows. Given any $\epsilon>0$, there is $\delta=\delta(\epsilon)>0$ such that for any $\delta$-cycle $(z_{i})_{i=0}^{k}$ of $f$, $d(z_{i},CR(f))\leq\epsilon$ holds for all $0\leq i\leq k$. Fix $x,y\in CR(f)$ and let $\alpha=(x_{i})_{i=0}^{l}$, $\beta=(y_{i})_{i=0}^{l}\subset X$ be $\delta$-chains of $f$ with $x_{0}=y_{l}=x$ and $x_{l}=y_{0}=y$. Then, since $\gamma=\alpha\beta$ is a $\delta$-cycle of $f$, the above property holds. Put $x^{\prime}_{0}=x$, $x^{\prime}_{l}=y$, and take $x^{\prime}_{i}\in CR(f)$ with $d(x_{i},x^{\prime}_{i})=d(x_{i},CR(f))\leq\epsilon$ for each $0<i<l$. Then, for any $\delta^{\prime}>0$, if $\delta$ is sufficiently small, $\alpha^{\prime}=(x^{\prime}_{i})_{i=0}^{l}\subset CR(f)$ gives a $\delta^{\prime}$-chain of $f$, which has the same length and end points as $\alpha$. This argument also applies to $\beta$. We refer to the Robinson’s proof of $CR(f|_{CR(f)})=CR(f)$ in [13] for a similar argument.

Then, the basic properties of the relation $\sim$ are given in the following lemma.

Lemma 3.1**.**

For any continuous map $f:X\to X$, the relation $\sim$ on $CR(f)$ is a closed $(f\times f)$-invariant equivalence relation. In other words, $\sim$ is an equivalence relation on $CR(f)$, $R=\{(x,y)\in CR(f)^{2}:x\sim y\}$ is a closed subset of $CR(f)^{2}$, and $(f\times f)(R)\subset R$.

Remark 3.2**.**

Actually, we have $(f\times f)(R)=R$ in the above notation.

It is not difficult to give a direct proof of Lemma 3.1 based only on the definition of the relation $\sim$. However, we confirm it through an alternative description of $\sim$. In [5], it was shown that for any continuous map $f:X\to X$, $CR(f)$ admits the so-called $\delta$-cyclic decomposition for each $\delta>0$, that is, a family of disjoint subsets of $CR(f)$

[TABLE]

with the following properties, where $J=\{(i,j):1\leq i\leq K,\>0\leq j\leq m_{i}-1\}$.

• (D1)

$CR(f)=\bigsqcup_{(i,j)\in J}D_{i,j}$, and every $D_{i,j}$, $(i,j)\in J$, is clopen in $CR(f)$.

• (D2)

Putting $D_{i,m_{i}}=D_{i,0}$, we have $f(D_{i,j})=D_{i,j+1}$ for every $(i,j)\in J$.

• (D3)

Given any $x,y\in D_{i,j}$ with $(i,j)\in J$, there is $N>0$ such that for any integer $n\geq N$, there is a $\delta$-chain $(x_{\eta})_{\eta=0}^{k}\subset CR(f)$ of $f$ with $x_{0}=x$, $x_{k}=y$, and $k=m_{i}n$.

Then, Lemma 3.1 is an immediate consequence of the above properties (D1), (D2), and the following lemma.

Lemma 3.2**.**

Let $f:X\to X$ be a continuous map. Then, for any $x,y\in CR(f)$, $x\sim y$ if and only if for every $\delta>0$, $x$ and $y$ are contained in the same component of $\mathcal{D}(\delta)$.

Proof.

Suppose $x\sim y$. For any given $\delta>0$, let $\mathcal{D}(\delta)=\{D_{i,j}:1\leq i\leq K,\>0\leq j\leq m_{i}-1\}$ be the $\delta$-cyclic decomposition of $CR(f)$. Without loss of generality, we may assume $x\in D_{1,0}$. Then, it must be shown that $y\in D_{1,0}$. For the purpose, we take $\delta^{\prime}>0$ such that $d(A,B)=\inf\{d(a,b):a\in A,b\in B\}>\delta^{\prime}$ for any $A,B\in\mathcal{D}(\delta)$ with $A\neq B$. For any integer $n=qm_{1}+r\geq 0$, where $q\geq 0$ and $0\leq r\leq m_{1}-1$, put $D_{1,n}=D_{1,r}$. Then, for every $\delta^{\prime}$-chain $(x_{n})_{n=0}^{L}\subset CR(f)$ of $f$ with $x_{0}=x$, by (D2) and the choice of $\delta^{\prime}$, we have $x_{n}\in D_{1,n}$ for all $0\leq n\leq L$. Now, since $x\sim y$, there are $m=m(\delta^{\prime})>0$ and $N=N(\delta^{\prime})>0$ as in Definition 3.1. Put $M=mm_{1}N$ and note that $m_{1}N\geq N$. Thus, there is a $\delta^{\prime}$-chain $(x_{n})_{n=0}^{M}\subset CR(f)$ of $f$ such that $x_{0}=x$ and $x_{M}=y$, and this implies $y=x_{(mN)m_{1}}\in D_{1,(mN)m_{1}}=D_{1,0}$. The converse is a direct consequence of (D3). ∎

The following lemma relates the entropy pairs with the chain relation $\sim$.

Lemma 3.3**.**

Let $f:X\to X$ be a continuous map. Then, for any $(x,y)\in E(X,f)$, $(x,y)\in CR(f)^{2}$ and $x\sim y$.

Proof.

Suppose $(x,y)\in E(X,f)$. Given any $\delta>0$, let $\mathcal{D}(\delta)$ be the $\delta$-cyclic decomposition of $CR(f)$. We define a relation $R\subset X^{2}$ by

[TABLE]

Then, by (D1) and (D2), $R$ is a closed $(f\times f)$-invariant equivalence relation. Let $X_{R}$ be the quotient space, $\pi:X\to X_{R}$ be the quotient map, and $f_{R}:X_{R}\to X_{R}$ be the continuous map defined by $f_{R}\circ\pi=\pi\circ f$. $\pi$ is a factor map. We easily see that $\Omega(f_{R})=\{\pi(A):A\in\mathcal{D}(\delta)\}$, a finite set (here $\Omega(\cdot)$ is the non-wandering set), and hence $h_{top}(f_{R})=h_{top}(f_{R}|_{\Omega(f_{R})})=0$. From Lemma 2.1, it follows that $\pi(x)=\pi(y)$ (or $(x,y)\in R$), which implies $\{x,y\}\subset A$ for some $A\in\mathcal{D}(\delta)$. Since $\delta>0$ is arbitrary, by Lemma 3.2, we conclude that $(x,y)\in CR(f)^{2}$ and $x\sim y$. ∎

Remark 3.3**.**

For any $(x,y)\in CR(f)^{2}$, if $x\sim y$, then we easily see that the following holds: Given any $\delta>0$, there are an integer $a>0$ and $\delta$-chains $\gamma_{ij}=(y_{ij,\eta})_{\eta=0}^{a}$ of $f$ with $y_{ij,0}=i$ and $y_{ij,a}=j$ for all $i,j\in\{x,y\}$ (P). Conversely, it is obvious that if (P) holds for $(x,y)\in CR(f)^{2}$, then $x\sim y$. Keeping this equivalence in mind, Lemma 3.3 can also be proved as below by Kerr and Li characterization of the entropy pairs as the so-called (non-diagonal) IE-pairs (see [7] for details).

Proof.

Let $(x,y)\in E(X,f)$. Then, according to [7], $(x,y)$ is an IE-pair for $f$. It follows that for any $\epsilon>0$, $(B_{\epsilon}(x),B_{\epsilon}(y))$ has an independence set of positive density (here $B_{\epsilon}(\cdot)$ is the $\epsilon$-ball). In particular, this implies the existence of two integers $0\leq m<n$ such that for all $i,j\in\{x,y\}$, there is $x_{ij}\in X$ with $f^{m}(x_{ij})\in B_{\epsilon}(i)$ and $f^{n}(x_{ij})\in B_{\epsilon}(j)$. Then, it is clear that the property (P) holds for $(x,y)$, thus $(x,y)\in CR(f)^{2}$ and $x\sim y$. ∎

4. Proof of Theorem 1.2

In this section, we prove Theorem 1.2. For the purpose, we first define a separation property for two chain recurrent points.

Definition 4.1**.**

Given a continuous map $f:X\to X$, we say that a pair of points $(x,y)\in CR(f)^{2}$ has property if there is $r>0$ such that for any $\delta>0$, there are two $\delta$-cycles $\gamma_{1}=(x_{i})_{i=0}^{k}$, $\gamma_{2}=(y_{i})_{i=0}^{k}\subset CR(f)$ of $f$ such that $x_{0}=x_{k}=x$, $y_{0}=y_{k}=y$, and $d(x_{i},y_{i})>r$ for each $0\leq i\leq k$.*

Let $f:X\to X$ be a continuous map. A pair of points $(x,y)\in X^{2}$ is said to be a proximal pair for $f$ if $\liminf_{n\to\infty}d(f^{n}(x),f^{n}(y))=0$. If $(x,y)\in X^{2}$ is not proximal, i.e. $\inf_{n\geq 0}d(f^{n}(x),f^{n}(y))>0$, the pair is said to be distal.

Remark 4.1**.**

(1) If $(x,y)\in CR(f)^{2}$ has property*, then $(x,y)$ is a distal pair for $f$.

(2) If $(x,y)\in X^{2}$ is a distal pair for $f$, then any $(z,w)\in\omega((x,y),f\times f)\subset CR(f)^{2}$ satisfies property*.

(3) Assume that $(x,y)\in CR(f)^{2}$ with $x\sim y$ is a distal pair for $f$. Then, given any $(z,w)\in\omega((x,y),f\times f)\subset CR(f)^{2}$, since $\sim$ is a closed $(f\times f)$-invariant relation (by Lemma 3.1), $z\sim w$. It also holds that $(z,w)$ has property*.

The following lemma gives a sufficient condition for DC1.

Lemma 4.1**.**

Suppose that a continuous map $f:X\to X$ satisfies the following properties.

• (1)

There is $(z,w)\in CR(f)^{2}$ which satisfies $z\sim w$ and property.*

• (2)

For any sequence $(x_{i})_{i\geq 0}$ of points in $CR(f)$, if $\lim_{i\to\infty}d(f(x_{i}),x_{i+1})=0$, then there is $x\in X$ such that $\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}d(f^{i}(x),x_{i})=0$.

Then, $f$ exhibits distributional chaos of type 1 (DC1).

Proof.

By (1), there is $r>0$ with the following property: For any integer $n\geq 1$, there are an integer $a_{n}>0$, two $n^{-1}$-cycles $\gamma_{0,n}=(p_{i})_{i=0}^{a_{n}}$, $\gamma_{1,n}=(q_{i})_{i=0}^{a_{n}}\subset CR(f)$ of $f$, and two $n^{-1}$-chains $\alpha_{n}=(r_{i})_{i=0}^{a_{n}}$, $\beta_{n}=(s_{i})_{i=0}^{a_{n}}\subset CR(f)$ of $f$ such that $p_{0}=p_{a_{n}}=z$, $q_{0}=q_{a_{n}}=w$, $r_{0}=s_{a_{n}}=z$, $r_{a_{n}}=s_{0}=w$, and $d(p_{i},q_{i})>r$ for every $0\leq i\leq a_{n}$. We take a sequence of integers $0<m_{1}<m_{2}<\cdots$ such that $m_{1}=2$, and, putting $b_{n}=\sum_{i=0}^{n-1}a_{i}m_{i}$, we have

[TABLE]

for every $n>1$. Put $c_{0,n}=\gamma_{0,n}^{m_{n}}$ and $c_{1,n}=\alpha_{n}\gamma_{1,n}^{m_{n}-2}\beta_{n}$ for each $n\geq 1$. We have $l(c_{0,n})=l(c_{1,n})=a_{n}m_{n}$ for any $n\geq 1$ (here $l(\cdot)$ denotes the length of the cycle). Then, for each $u=(u_{n})_{n\in\mathbb{N}}\in\{0,1\}^{\mathbb{N}}$, define $\xi(u)=c_{u_{1},1}c_{u_{2},2}c_{u_{3},3}\cdots\subset CR(f)$, a limit pseudo orbit of $f$, and take $x(u)$ as in the property (2). Let us fix an uncountable subset $S\subset\{0,1\}^{\mathbb{N}}$ such that for any $u,v\in S$ with $u\neq v$, both $\{n\in\mathbb{N}:u_{n}=v_{n}\}$ and $\{n\in\mathbb{N}:u_{n}\neq v_{n}\}$ are infinite sets. We shall prove that for any $u,v\in S$ with $u\neq v$, $(x(u),x(v))\in X^{2}$ is a DC1-pair for $f$.

For the purpose, put $\xi(u)=(x_{0,i})_{i\geq 0}$, $\xi(v)=(x_{1,i})_{i\geq 0}$, and $(x(u),x(v))=(x_{0},x_{1})$. Note that, putting $\epsilon_{j,n}=\frac{1}{n}\sum_{i=0}^{n-1}d(f^{i}(x_{j}),x_{j,i})$, we have $\lim_{n\to\infty}\epsilon_{j,n}=0$ for each $j\in\{0,1\}$. Given any $\delta>0$, putting $c_{n}=b_{n}+a_{n}m_{n}+1$, we have

[TABLE]

for any $n\geq 1$ and $j\in\{0,1\}$. On the other hand, for any $n>1$, if $u_{n}=v_{n}$, then because $c_{u_{n},n}=c_{v_{n},n}$,

[TABLE]

and so

[TABLE]

Hence, for any $n>1$ with $u_{n}=v_{n}$, we have

[TABLE]

This together with $|\{n\in\mathbb{N}:u_{n}=v_{n}\}|=\infty$ yields

[TABLE]

Note that $\delta>0$ is arbitrary. It only remains to prove the following:

[TABLE]

Similarly as above, we have

[TABLE]

for any $n\geq 1$ and $j\in\{0,1\}$. On the other hand, for any $n>1$, if $u_{n}\neq v_{n}$, then because $c_{u_{n},n}\neq c_{v_{n},n}$,

[TABLE]

and so

[TABLE]

Thus, for any $n>1$ with $u_{n}\neq v_{n}$, we have

[TABLE]

This together with $|\{n\in\mathbb{N}:u_{n}\neq v_{n}\}|=\infty$ implies the required equation. Now, It has been proved that for any $u,v\in S$ with $u\neq v$, $(x(u),x(v))$ is a DC1-pair for $f$. Since $S$ is an uncountable set, $f$ exhibits DC1. ∎

We need the following lemma from [6].

Lemma 4.2**.**

[6, Theorem 1.1]** Let $f:X\to X$ be a continuous map with the limit shadowing property. Then, $f|_{CR(f)}:CR(f)\to CR(f)$ satisfies the shadowing property.

It is known that the shadowing property with positive topological entropy allows us to obtain a subsystem of $(X,f^{a})$ (for some $a>0$) which has $(\{0,1\}^{\mathbb{N}},\sigma)$ (here $\sigma$ is the shift map) as a factor (see, for example, [9, 11]). We use it and Lemma 3.3 to find a pair of chain recurrent points with the two properties assumed in Lemma 4.1. Since, especially, the second part of the proof of Theorem 1.2 relies on a specific construction of the subsystem through $\sim$, for completeness, we briefly describe it below.

Construction: Suppose that $g=f|_{CR(f)}:CR(f)\to CR(f)$ has the shadowing property. Given $(x,y)\in CR(f)^{2}$ with $x\neq y$ and $x\sim y$, put $x_{0}=x$, $x_{1}=y$, and let $0<\epsilon<2^{-1}d(x,y)$. Take $\delta=\delta(\epsilon)>0$ such that every $\delta$-pseudo orbit of $g$ is $\epsilon$-shadowed by some point of $CR(f)$. Since $x\sim y$, there are an integer $a>0$ and $\delta$-chains $\gamma_{ij}=(y_{ij,\eta})_{\eta=0}^{a}\subset CR(f)$ of $g$ with $y_{ij,0}=x_{i}$ and $y_{ij,a}=x_{j}$ for each $i,j\in\{0,1\}$. For any $s=(s_{i})_{i\in\mathbb{N}}\in\{0,1\}^{\mathbb{N}}$, consider the $\delta$-pseudo orbit $\xi(s)=\gamma_{s_{1}s_{2}}\gamma_{s_{2}s_{3}}\gamma_{s_{3}s_{4}}\cdots\subset CR(f)$ of $g$, which is $\epsilon$-shadowed by some $x(s)\in CR(f)$. Let

[TABLE]

and note that $Y$ is a compact $f^{a}$-invariant subset of $CR(f)$. Define a map $\pi:Y\to\{0,1\}^{\mathbb{N}}$ by the condition that $\xi(\pi(x))$ is $\epsilon$-shadowed by $x$ for each $x\in Y$. Then, it is easy to see that $\pi:(Y,f^{a})\to(\{0,1\}^{\mathbb{N}},\sigma)$ is a factor map.

Remark 4.2**.**

By the above construction, we see that if $f|_{CR(f)}$ has the shadowing property, then for any $(x,y)\in CR(f)^{2}$ with $x\neq y$, $x\sim y$ implies $(x,y)\in E(X,f)$. Indeed, we have $x(s)\in\bigcap_{i\in\mathbb{N}}f^{-(i-1)a}(B_{\epsilon}(x_{s_{i}}))\neq\emptyset$ for every $s\in\{0,1\}^{\mathbb{N}}$ in the above notation, which implies

[TABLE]

Since $0<\epsilon<2^{-1}d(x,y)$ is arbitrary, $(x,y)\in E(X,f)$. The fact that the shift map $\sigma:\{0,1\}^{\mathbb{N}}\to\{0,1\}^{\mathbb{N}}$ has upe, which is mentioned in Section 2, is a consequence of this remark.

Combining this remark with Lemma 3.3 (and Lemma 2.1), we get the following corollary, which characterizes the set of entropy pairs by the relation $\sim$, under the assumption of the shadowing property.

Corollary 4.1**.**

Let $f:X\to X$ be a continuous map. If $f|_{CR(f)}:CR(f)\to CR(f)$ has the shadowing property, then it holds that

[TABLE]

In particular, $h_{top}(f)>0$ if only if $x\sim y$ for some $(x,y)\in CR(f)^{2}$ with $x\neq y$.

Finally, let us prove Theorem 1.2.

Proof of Theorem 1.2.

By (1) and Lemma 4.2, $f|_{CR(f)}:CR(f)\to CR(f)$ satisfies the shadowing property. Since $E(X,f)\neq\emptyset$ by (2) and Lemma 2.1, we can take $(x,y)\in E(X,f)$. Then, by Lemma 3.3, it holds that $(x,y)\in CR(f)^{2}$, $x\neq y$, and $x\sim y$; therefore, we have a factor map $\pi:(Y,f^{a})\to(\{0,1\}^{\mathbb{N}},\sigma)$ as above. Since $\sigma$ has upe, especially, we have $(0^{\infty},1^{\infty})\in E(\{0,1\}^{\mathbb{N}},\sigma)$. According to Lemma 2.1, there is $(p,q)\in E(Y,f^{a})$ such that $\pi(p)=0^{\infty}$ and $\pi(q)=1^{\infty}$. Because $(0^{\infty},1^{\infty})$ is a distal pair for $\sigma$, $(p,q)$ is a distal pair for $f^{a}$, so for $f$. On the other hand, since

[TABLE]

we have $(p,q)\in E(X,f)$, which together with Lemma 3.3 implies $(p,q)\in CR(f)^{2}$ and $p\sim q$. Fix $(z,w)\in\omega((p,q),f\times f)\subset CR(f)^{2}$. Then, as Remark 4.1, it satisfies $z\sim w$ and property*. Thus, by (1) and Lemma 4.1, we conclude that $f$ exhibits DC1.

Let us prove $E(X,f)\subset\overline{DC1(X,f)}$. Given any $(x,y)\in E(X,f)$, Lemma 3.3 implies $(x,y)\in CR(f)^{2}$, $x\neq y$, and $x\sim y$, so we have a factor map $\pi:(Y,f^{a})\to(\{0,1\}^{\mathbb{N}},\sigma)$ as above (for any $0<\epsilon<2^{-1}d(x,y)$). Take $(p,q)\in E(Y,f^{a})$ such that $\pi(p)=0^{\infty}$ and $\pi(q)=1^{\infty}$. Fix $(z,w)\in\omega((p,q),f^{a}\times f^{a})$ and note that

[TABLE]

Since $\omega((p,q),f^{a}\times f^{a})\subset\omega((p,q),f\times f)$, similarly as above, $(z,w)\in CR(f)^{2}$ satisfies $z\sim w$ and property*. Then, we see that, because $f$ has the limit shadowing property, in the proof of Lemma 4.1, $x(u)$ can be taken as a limit shadowing point of $\xi(u)$ for any $u\in\{0,1\}^{\mathbb{N}}$. Also, in the proof of Lemma 4.1, it has been proved that for any $u,v\in\{0,1\}^{\mathbb{N}}$ with $|\{n\in\mathbb{N}:u_{n}=v_{n}\}|=\infty$ and $|\{n\in\mathbb{N}:u_{n}\neq v_{n}\}|=\infty$, $(x(u),x(v))$ is a DC1-pair for $f$. For concreteness, put $u=0^{\infty}$ and $v=(01)^{\infty}=0101\cdots$. Then, by the definition of $\xi(u)$ and $\xi(v)$, we easily see that there is a sequence of integers $0<n_{1}<n_{2}\cdots$ such that

[TABLE]

Since every $(f^{n_{j}}(x(u)),f^{n_{j}}(x(v)))$, $j\geq 1$, is a DC1-pair for $f$ as $(x(u),x(v))$ is so, we have $(z,w)\in\overline{DC1(X,f)}$. Thus, by $(z,w)\in B_{\epsilon}(x)\times B_{\epsilon}(y)$, and since $0<\epsilon<2^{-1}d(x,y)$ is arbitrary, $(x,y)\in\overline{DC1(X,f)}$, completing the proof. ∎

Appendix A

In this appendix, as a complement to the proof of Theorem 1.2, we show that the existence of $(z,w)\in CR(f)^{2}$ with $z\sim w$ and property* is a necessary condition for DC1. For the purpose, the next lemma is needed.

Lemma A.1**.**

Let $f:X\to X$ be a continuous map. If $(x,y)\in X^{2}$ is a proximal pair for $f$, then any $(z,w)\in\omega((x,y),f\times f)\subset CR(f)^{2}$ satisfies $z\sim w$.

Proof.

Given any $\delta>0$, consider the $\delta$-cyclic decomposition $\mathcal{D}(\delta)$ of $CR(f)$. For any $p\in X$, note that $\lim_{n\to\infty}d(f^{n}(p),CR(f))=0$, and so by (D2), we have $\lim_{n\to\infty}$ $d(f^{n}(p),f^{n}(A_{p}))=0$ for some $A_{p}\in\mathcal{D}(\delta)$. It holds that

[TABLE]

Since $(x,y)$ is proximal, again by (D2), $A_{x}$ and $A_{y}$ should be equal. Put $A=A_{x}=A_{y}$. Then, because $(z,w)\in\omega((x,y),f\times f)$, there are $B\in\mathcal{D}(\delta)$ and a sequence of integers $0<n_{1}<n_{2}<\cdots$ such that $f^{n_{i}}(A)=B$ for all $i\geq 1$, and $\lim_{i\to\infty}(f^{n_{i}}(x),f^{n_{i}}(y))=(z,w)$. These properties give $\{z,w\}\subset B$, and since $\delta>0$ is arbitrary, by Lemma 3.2, we conclude $z\sim w$. ∎

Proposition A.1**.**

Let $f:X\to X$ be a continuous map. If $(x,y)\in DC1(X,f)$, then there exists $(z,w)\in\omega((x,y),f\times f)\subset CR(f)^{2}$ with $z\sim w$ and property.*

Proof.

Let $(x,y)\in DC1(X,f)$. Then, $(x,y)$ is clearly a proximal pair for $f$. From the definition of DC1-pairs, it also follows that there is $\delta_{0}>0$ for which

[TABLE]

is a so-called thick set, meaning that for any $n>0$, there is $j\geq 0$ such that $\{j,j+1,\dots,j+n\}\subset T$. Let us choose two sequences of integers $0<n_{1}<n_{2}<\cdots$, $0<N_{1}<N_{2}<\cdots$, and $(p,q)\in X^{2}$ such that the following holds.

• (1)

For any $j\geq 1$ and any $n_{j}\leq n\leq n_{j}+N_{j}$, $d(f^{n}(x),f^{n}(y))>\delta_{0}$.

• (2)

$\lim_{j\to\infty}(f^{n_{j}}(x),f^{n_{j}}(y))=(p,q)$.

Then, $d(f^{n}(p),f^{n}(q))\geq\delta_{0}$ for all $n\geq 0$; therefore, $(p,q)\in\omega((x,y),f\times f)$ is distal. By Lemma A.1, it also holds that $p\sim q$. Take $(z,w)\in\omega((p,q),f\times f)\subset\omega((x,y),f\times f)$. Then, as Remark 4.1, $z\sim w$ and property* hold. ∎

We end with a simple application of Lemma A.1. A pair of points $(x,y)\in X^{2}$ is said to be a Li-Yorke pair for $f$ if

[TABLE]

By Lemma A.1, a similar argument as the proof of Proposition A.1 shows that for any Li-Yorke pair $(x,y)$ for $f$, there is $(z,w)\in\omega((x,y),f\times f)\subset CR(f)^{2}$ such that $z\neq w$ and $z\sim w$. When $f|_{CR(f)}:CR(f)\to CR(f)$ has the shadowing property, due to Corollary 4.1, the existence of such a $(z,w)$ implies $h_{top}(f)>0$. Thus, by Lemma 4.2, the assumption (2) of Theorem 1.2 holds if $f$ has the limit shadowing property and has a Li-Yorke pair. Also, it follows that a continuous map $f$ with the limit shadowing property exhibits DC1 only if $h_{top}(f)>0$ because any DC1-pair is a Li-Yorke pair.