Topological conjugacy between induced non-autonomous set-valued systems and subshifts of finite type

TL;DR

This paper explores the conditions under which non-autonomous set-valued systems are topologically conjugate or semiconjugate to subshifts of finite type, extending existing results and providing criteria for chaos and entropy estimation.

Contribution

It establishes new conditions for conjugacy and semiconjugacy between non-autonomous systems and subshifts, extending autonomous system results to non-autonomous cases.

Findings

Derived conditions for topological conjugacy and semiconjugacy.

Provided entropy estimates and chaos criteria.

Extended results from autonomous to non-autonomous systems.

Abstract

This paper establishes topological (equi-)semiconjugacy and (equi-)conjugacy between induced non-autonomous set-valued systems and subshifts of finite type. First, some necessary and sufficient conditions are given for a non-autonomous discrete system to be topologically semiconjugate or conjugate to a subshift of finite type. Further, several sufficient conditions for it to be topologically equi-semiconjugate or equi-conjugate to a subshift of finite type are obtained. Consequently, estimations of topological entropy and several criteria of Li-Yorke chaos and distributional chaos in a sequence are derived. Second, the relationships of several related dynamical behaviors between the non-autonomous discrete system and its induced set-valued system are investigated. Based on these results, the paper furthermore establishes the topological (equi-)semiconjugacy and (equi-)conjugacy between induced set-valued systems and subshifts of finite type. Consequently, estimations of the topological entropy for the induced set-valued system are obtained, and several criteria of Li-Yorke chaos and distributional chaos in a sequence are established. Some of these results not only extend the existing related results for autonomous discrete systems to non-autonomous discrete systems, but also relax the assumptions of the counterparts in the literature. Two examples are finally provided for illustration.

Equations

$$x_{n+1}=f_n(x_n),n\ge 0,$$

$$\underset{n\to \infty }{\mathrm{liminf}}d(f_0^n(x),f_0^n(y))=0\mathrm{and}\underset{n\to \infty }{\mathrm{limsup}}d(f_0^n(x),f_0^n(y))>\delta .\text{vspace}-0.2cm$$

$$h(f_{0,\infty },Y):=\underset{\mathcal{A}}{\sup }\left\{\underset{n\to \infty }{\mathrm{limsup}}\log \mathcal{N}({\mathcal{A}}_0^n{|}_Y)/n\right\}.\text{vspace}-0.2cm$$

$$u_{n+1}=g_n(u_n),n\ge 0,$$

$$d(x_n,y_n)>\frac{ϵ}{2},n\ge N.$$

$$\mathcal{H}_{d}(A,B):=\max\bigg{\{}\sup_{a\in A}d(a,B),\sup_{b\in B}d(b,A)\bigg{\}},$$

$$⟨K⟩:=\{A\in \mathcal{K}(X):A\subset K\}.$$

$$A_{n+1}={\bar{f}}_n(A_n),n\ge 0,$$

$${\Sigma }_N^{+}(A):=\{s=(s_0,s_1,\cdots ):1\le s_j\le N,a_{s_j{s}_{j+1}}=1,j\ge 0\}.\text{vspace}-0.2cm$$

$$\hat{\rho }(\alpha ,\beta ):=\sum _{i=0}^{\infty }\hat{d}(a_i,b_i)/2^i,\alpha =(a_0,a_1,\cdots ),\beta =(b_0,b_1,\cdots )\in {\Sigma }_N^{+}(A),\text{vspace}-0.2cm$$

$$\rho (A):=\underset{n\to \infty }{\lim }\Vert A^n{\Vert }^{\frac{1}{n}},\Vert A\Vert =\sum _{1\le i,j\le N}{a}_{ij}.$$

$$f_n(V_{i,n})\supset \bigcup _{a_{ij}=1}{V}_{j,n+1},1\le i\le N,n\ge 0,\text{vspace}-0.2cm$$

$$U_i:=\{\alpha =(a_0,a_1,\cdots )\in {\Sigma }_N^{+}(A):a_0=i\},1\le i\le N.$$

$$\bigcup _{i=1}^N{U}_i={\Sigma }_N^{+}(A).$$

$$E:=\bigcup _{n=0}^{\infty }\bigcup _{i=1}^N{V}_{i,n},V_{\alpha }^{m,n}:=\bigcap _{k=0}^m{f}_n^{-k}(V_{a_k,n+k}).$$

$$d(f_n^m(x),f_n^m(y))\ge {\lambda }^{m}d(x,y),\text{vspace}-0.2cm$$

$$d(x,y)\le {\lambda }^{-m}d(f_n^m(x),f_n^m(y))\le {\lambda }^{-m}\underset{1\le i\le N}{\max }d(V_i).\text{vspace}-0.2cm$$

$$d(V_{\alpha }^{m,n})\le {\lambda }^{-m}\underset{1\le i\le N}{\max }d(V_i).\text{vspace}-0.2cm$$

$${\Lambda }_n:=\bigcup _{\alpha \in {\Sigma }_N^{+}(A)}\bigcap _{m=0}^{\infty }{V}_{\alpha }^{m,n}.$$

$$d(f_n^j(x),f_n^j(y))<\underset{1\le i_1\ne i_2\le N}{\min }d(V_{i_1,n+j},V_{i_2,n+j}),0\le j\le N_1.\text{vspace}-0.2cm$$

$$\hat{\rho }({\phi }_n(x),{\phi }_n(y))=\hat{\rho }(\alpha ,\beta )\le 2^{-N_1}<ϵ.\text{vspace}-0.2cm$$

$${\hat{\phi }}_{n+1}\circ f_n={\sigma }_A\circ {\hat{\phi }}_n,n\ge 0.$$

$${\hat{\phi }}_{n+k}\circ f_n^k={\sigma }_A^k\circ {\hat{\phi }}_n,n\ge 0,k\ge 1.$$

$$V_{i,n}:={\hat{\phi }}_n^{-1}(U_i),1\le i\le N,n\ge 0,$$

$$V_{\alpha}^{m,n}=\bigcap_{k=0}^{m}f_{n}^{-k}(V_{a_{k},n+k})=\bigcap_{k=0}^{m}(f_{n}^{-k}\circ\hat{\varphi}_{n+k}^{-1})(U_{a_{k}})=\bigcap_{k=0}^{m}(\hat{\varphi}_{n}^{-1}\circ\sigma_{A}^{-k})(U_{a_{k}})=\hat{\varphi}_{n}^{-1}\bigg{(}\bigcap_{k=0}^{m}\sigma_{A}^{-k}(U_{a_{k}})\bigg{)}.\vspace{-0.2cm}$$

$$\bigcap _{m=0}^{\infty }{V}_{\alpha }^{m,n}:=\{x^n(\alpha )\},{\Lambda }_n:=\bigcup _{\alpha \in {\Sigma }_N^{+}(A)}\{x^n(\alpha )\}.\text{vspace}-0.2cm$$

$$f_n(x^n(\alpha ))=x^{n+1}({\sigma }_A(\alpha )).$$

$$h_{n+1}\circ {\sigma }_A(\alpha )=x^{n+1}({\sigma }_A(\alpha ))=f_n(x^n(\alpha ))=f_n\circ h_n(\alpha ),\text{vspace}-0.2cm$$

$$h_{n+1}\circ {\sigma }_A=f_n\circ h_n.$$

$$d(h_n(\alpha ),h_n(\beta ))=d(x^n(\alpha ),x^n(\beta ))\le d(V_{\alpha }^{N_2,n})<ϵ.\text{vspace}-0.2cm$$

Full text

**Topological conjugacy between induced non-autonomous set-valued systems and subshifts of finite type

**

Hua Shao*†, Guanrong Chen†, Yuming Shi‡*

† Department of Electronic Engineering, City University of Hong Kong,

Hong Kong SAR, P. R. China

‡ Department of Mathematics, Shandong University

Jinan, Shandong 250100, P. R. China

‡ The corresponding author: ymshi@sdu.edu.cn

Abstract. This paper establishes topological (equi-)semiconjugacy and (equi-)conjugacy between induced non-autonomous set-valued systems and subshifts of finite type. First, some necessary and sufficient conditions are given for a non-autonomous discrete system to be topologically semiconjugate or conjugate to a subshift of finite type. Further, several sufficient conditions for it to be topologically equi-semiconjugate or equi-conjugate to a subshift of finite type are obtained. Consequently, estimations of topological entropy and several criteria of Li-Yorke chaos and distributional chaos in a sequence are derived. Second, the relationships of several related dynamical behaviors between the non-autonomous discrete system and its induced set-valued system are investigated. Based on these results, the paper furthermore establishes the topological (equi-)semiconjugacy and (equi-)conjugacy between induced set-valued systems and subshifts of finite type. Consequently, estimations of the topological entropy for the induced set-valued system are obtained, and several criteria of Li-Yorke chaos and distributional chaos in a sequence are established. Some of these results not only extend the existing related results for autonomous discrete systems to non-autonomous discrete systems, but also relax the assumptions of the counterparts in the literature. Two examples are finally provided for illustration.

Keywords: non-autonomous discrete system; induced set-valued system; subshift of finite type; topological conjugacy; topological entropy; chaos.

2010 Mathematics Subject Classification: 37B55, 54C60, 37B10.

1. Introduction

Symbolic dynamical systems play a significant role in the study of chaos theory of dynamical systems since they appear to be simple but have quite rich and complex dynamical behaviors. They were first utilized by Hadamard to study geodesics on surfaces with negative curvatures [7]. Then, they were applied by Birkhoff to study dynamical systems [1]. In the 1960s, complex dynamical behaviors of Smale horseshoe were depicted by its topological conjugacy to a two-sided symbolic system [31]. Two well-known maps, the Hénon map and the logistic map, were shown to be chaotic for some parameters also by their topological conjugacy to the symbolic dynamical systems [3, 4, 19]. In fact, it is a very useful method for studying the complexity of dynamical systems by establishing topological semiconjugacy or conjugacy to symbolic dynamical systems (see [2, 11, 27, 28, 29, 30, 37], and references therein).

It is worth mentioning that Block and Coppel in 1992 introduced the concept of “turbulence” for continuous interval maps, and proved that a strictly turbulent map is topologically semi-conjugate to the one-sided symbolic system on two symbols in a compact invariant set [2]. In 2004, we proved that a strictly turbulent map, which satisfies an expanding condition in distance, is topologically conjugate to the one-sided symbolic system on two symbols [27]. In 2006, we changed the term “turbulence” to “coupled-expansion” in order to avoid possible confusion with the turbulence concept in fluid mechanics [28]. Later, we extended this notion to coupled-expansion for a transition matrix and showed that a map, which is strictly coupled-expanding for a transition matrix, is topological conjugate to a subshift of finite type under certain conditions [30]. Thereafter, chaos induced by coupled-expansion for a transition matrix has attracted interest and attention from researchers in the field [9, 12, 14, 22, 29, 35, 36, 37].

In the study of complex dynamics, sometimes it is not enough to know only the tracjectory of a single point, but it needs to know the motion of a collection of points. For instance, in the study of collective behaviors in biology, one needs to know the massive migration of birds or mammals. Inspired by this natural phenomenon, Román-Flores studied the relationship between transitivity of a continuous map and its induced set-valued system [20]. Following his work, many scholars studied the relationships between individual chaos and collective chaos [5, 6, 10, 15, 16, 17, 21, 33, 34]. In particular, Wang and Wei proved that, if a continuous map is strictly coupled-expanding, then its induced set-valued discrete system is topologically semi-conjugate to a full shift [33]. Recently, Ju et al. generalized this result and gave some sufficient conditions under which the induced set-valued system is topologically (semi-)conjugate to a subshift of finite type [10].

Motivated by the above research, we are interested in studying the topological (equi-)semiconjugacy and (equi-)conjugacy between the induced non-autonomous set-valued system and a subshift of finite type. We achieve this goal in two steps. First, we establish the topological (equi-)semiconjugacy and (equi-)conjugacy between a non-autonomous discrete system and a subshift of finite type. Second, we investigate the relationships of several related dynamical behaviors between the non-autonomous discrete system and its induced set-valued system. Based on these results, we can establish the topological (equi-)semiconjugacy and (equi-)conjugacy between the induced non-autonomous set-valued system and a subshift of finite type. Since the complex dynamical behaviors of subshifts of finite type are well understood (see, for example, [38]), and the relationships of Li-Yorke chaos and topological entropy of two topological equi-(semi)conjugate non-autonomous discrete systems have been given in [29] and [13, 23], respectively, to that end several criteria of Li-Yorke chaos and distributional chaos in a sequence and estimations of the topological entropy for the induced set-valued system are obtained in this paper.

The rest of the present paper is organized as follows. Section 2 presents some basic concepts and useful lemmas. In Section 3, the topological (equi-)semiconjugacy and (equi-)conjugacy between a non-autonomous discrete system and a subshift of finite type are studied, and consequently estimations of topological entropy and some criteria of Li-Yorke chaos and distributional chaos in a sequence for the non-autonomous discrete system are established. In Section 4, the relationships of several related dynamical behaviors between the non-autonomous discrete system and its induced set-valued system are discussed, and the topological (equi-)semiconjugacy and (equi-)conjugacy between the induced non-autonomous set-valued system and a subshift of finite type are investigated. By applying the above-obtained results, estimations of the topological entropy for the induced non-autonomous set-valued system are obtained, and several criteria of Li-Yorke chaos and distributional chaos in a sequence are established. Two examples are finally provided in Section 5 for illustration.

2. Preliminaries

This section is divided into three parts for convenience of discussion. In Section 2.1, several basic concepts and related lemmas are introduced. Then, the notion of induced non-autonomous set-valued system is introduced in Section 2.2. In Section 2.3, the concepts of subshifts of finite type and weak coupled-expansion for transition matrices are reviewed.

2.1. Some basic concepts and related lemmas

Consider the following non-autonomous discrete system:

[TABLE]

where $(X,d)$ is a metric space and $f_{n}:X\to X$ is a map, $n\geq 0$. Denote $f_{0,\infty}:=\{f_{n}\}_{n=0}^{\infty}$, $f^{n}_{i}:=f_{i+n-1}\circ\cdots\circ f_{i}$, and $f^{-n}_{i}:=(f_{i}^{n})^{-1}$, $i\geq 0$, $n\geq 1$. Let $x\in X$ and $A,B$ be nonempty subsets of $X$. The boundary of $A$ is denoted by $\partial A$; the diameter of $A$ is denoted by $d(A)$; the distance between $x$ and $A$ is denoted by $d(x,A):=\inf\{d(x,a):a\in A\}$; and the distance between $A$ and $B$ is denoted by $d(A,B):=\inf\{d(a,b):a\in A,b\in B\}$. The set of all nonnegative integers and positive integers are denoted by $\mathbf{N}$ and $\mathbf{Z}^{+}$, respectively.

Definition 2.1 ([29], Definition 2.7). System (2.1) is said to be Li-Yorke $\delta$-chaotic for some $\delta>0$ if it has an uncountable Li-Yorke $\delta$-scrambled set $S$ in $X$; that is, for any $x,y\in S\subset X$,

[TABLE]

Further, it is said to be chaotic in the strong sense of Li-Yorke if all the orbits starting from the points in $S$ are bounded.

Definition 2.2 ([24], Definitions 2.1 and 2.2). System (2.1) is said to be distributionally chaotic if it has an uncountable distributionally scrambled set $D$ in $X$; that is, for any $x,y\in D\subset X$,

• (i)

\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}\chi_{[0,\epsilon)}\big{(}d(f_{0}^{i}(x),f_{0}^{i}(y))\big{)}=1 for any $\epsilon>0$,

• (ii)

\liminf_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}\chi_{[0,\delta)}\big{(}d(f_{0}^{i}(x),f_{0}^{i}(y))\big{)}=0 for some $\delta>0$,

where $\chi_{[0,\epsilon)}$ is the characteristic function defined on the set $[0,\epsilon)$.

The definition of topological entropy for system (2.1) is introduced in [13]. Let $X$ be compact, $Y$ be a nonempty subset of $X$, $\mathcal{A}$ be an open cover of $X$, and $\mathcal{N}(\mathcal{A})$ be the minimal possible cardinality of all subcovers chosen from $\mathcal{A}$. Denote the cover $\{A\cap Y:A\in\mathcal{A}\}$ of the set $Y$ by $\mathcal{A}|_{Y}$ and $\mathcal{A}_{0}^{n}:=\{\bigcap_{j=0}^{n-1}f_{0}^{-j}(A_{j}):A_{j}\in\mathcal{A},\;0\leq j\leq n-1\}$. Then the topological entropy of system (2.1) on $Y$ is defined by

[TABLE]

If $Y=X$, then $h(f_{0,\infty}):=h(f_{0,\infty},X)$ is called the topological entropy of system (2.1) on $X$.

Now, recall the definitions of topological (equi-)semiconjugacy and (equi-)conjugacy between system (2.1) and

[TABLE]

defined on a metric space $(Y,e)$, where $g_{n}:Y\to Y$ is a map, $n\geq 0$.

Let $\{\Lambda_{n}\}_{n=0}^{\infty}$ and $\{E_{n}\}_{n=0}^{\infty}$ be two sequences of subsets of $X$ and $Y$, respectively, and $h_{n}:\Lambda_{n}\to E_{n}$ be a map, $n\geq 0$. The sequence of maps $\{h_{n}\}_{n=0}^{\infty}$ is called equi-continuous in $\{\Lambda_{n}\}_{n=0}^{\infty}$ if for any $\epsilon>0$ there exists $\delta>0$ such that $e(h_{n}(x),h_{n}(y))<\epsilon$ for any $n\geq 0$ and any $x,y\in\Lambda_{n}$ with $d(x,y)<\delta$. In addition, $\{\Lambda_{n}\}_{n=0}^{\infty}$ is called invariant under system (2.1) if $f_{n}(\Lambda_{n})\subset\Lambda_{n+1}$ for any $n\geq 0$. Then, system (2.1) restricted to $\{\Lambda_{n}\}_{n=0}^{\infty}$ is an invariant subsystem of system (2.1) on $\{\Lambda_{n}\}_{n=0}^{\infty}$ [23].

Definition 2.3 ([29], Definition 3.3). Let $\{\Lambda_{n}\}_{n=0}^{\infty}$ and $\{E_{n}\}_{n=0}^{\infty}$ be invariant under systems (2.1) and (2.2), respectively. If for any $n\geq 0$, there exists an (equi-)continuous surjective map $h_{n}:\Lambda_{n}\to E_{n}$ such that $h_{n+1}\circ f_{n}=g_{n}\circ h_{n}$, then the invariant subsystem of system system (2.1) on $\{\Lambda_{n}\}_{n=0}^{\infty}$ is said to be topologically $\{h_{n}\}_{n=0}^{\infty}$ -(equi-)semiconjugate to the invariant subsystem of system (2.2) on $\{E_{n}\}_{n=0}^{\infty}$. Further, if $h_{n}$ is invertible for all $n\geq 0$ and $\{h_{n}^{-1}\}_{n=0}^{\infty}$ is also (equi-)continuous, then the above two invariant subsystems are said to be topologically $\{h_{n}\}_{n=0}^{\infty}$-(equi-)conjugate.

The relationships of Li-Yorke chaos and distributional chaos between two topological equi-conjugate systems are given below, respectively.

Lemma 2.1 ([26, 29]). Assume that the invariant subsystem of system (2.1) on $\{\Lambda_{n}\}_{n=0}^{\infty}$ is topologically equi-conjugate to the invariant subsystem of system (2.2) on $\{E_{n}\}_{n=0}^{\infty}$. Then,

• (i)

system (2.1) has an uncountable Li-Yorke $\delta$-scrambled set in $\Lambda_{0}$ if and only if system (2.2) has an uncountable Li-Yorke $\gamma$-scrambled set in $E_{0}$, for some $\delta,\gamma>0$;

• (ii)

system (2.1) has an uncountable distributionally scrambled set in $\Lambda_{0}$ if and only if system (2.2) has an uncountable distributionally scrambled set in $E_{0}$.

The next result shows the relationship of the topological entropy between two topological equi-(semi)conjugate systems.

Lemma 2.2 ([23], Lemma 2.2). Let $(X,d)$ and $(Y,e)$ be two compact metric spaces. If an invariant subsystem of system (2.1) on $\{\Lambda_{n}\}_{n=0}^{\infty}$ is topologically equi-semiconjugate to an invariant subsystem of system (2.2) on $\{E_{n}\}_{n=0}^{\infty}$, then $h(f_{0,\infty},\Lambda_{0})\geq h(g_{0,\infty},E_{0})$. Further, if they are topologically equi-conjugate, then $h(f_{0,\infty},\Lambda_{0})=h(g_{0,\infty},E_{0})$.

The following result will also be useful in the sequent sections.

Lemma 2.3. Let $\{E_{n}\}_{n=0}^{\infty}$ be a sequence of nonempty compact subsets of $X$ satisfying that $E_{n+1}\subset E_{n}$ for all $n\geq 0$. Then, $\bigcap_{n=0}^{\infty}{E_{n}}$ is a singleton if and only if $\lim_{n\to\infty}d(E_{n})=0$.

Proof. The sufficiency can be directly derived form Lemma 2.7 in [27]. To show the necessity, assume otherwise. Then, $\lim_{n\to\infty}d(E_{n})=\epsilon>0$. So, there exists $N\in\mathbf{Z}^{+}$ such that $d(E_{n})>\epsilon/2$ for all $n\geq N$. Thus, there exist $x_{n},y_{n}\in E_{n}$ satisfying that

[TABLE]

Fix any $n\geq 0$, and let $m\geq\max\{N,n\}$. Then, $\{x_{k}\}_{k=m}^{\infty},\{y_{k}\}_{k=m}^{\infty}\subset E_{m}\subset E_{n}$. Suppose that $x_{n}\to x$ and $y_{n}\to y$ as $n\to\infty$. Since $E_{n}$ is compact, $x,y\in E_{n}$. Thus, $x,y\in\bigcap_{n=0}^{\infty}E_{n}$. By (2.3) one has that $x\neq y$. This is a contradiction to the assumption that $\bigcap_{n=0}^{\infty}{E_{n}}$ is a singleton. Therefore, $\lim_{n\to\infty}d(E_{n})=0$. This completes the proof.

2.2. The induced non-autonomous set-valued system

Let $\mathcal{K}(X)$ be the class of all nonempty compact subsets of $X$ and the Hausdorff metric on $\mathcal{K}(X)$ be defined by

[TABLE]

where $A,B\in\mathcal{K}(X)$. It is known that $(\mathcal{K}(X),\mathcal{H}_{d})$ is a compact metric space if and only if $(X,d)$ is a compact metric space [18]. For convenience, let $\mathcal{H}_{d}(\mathcal{U})$ denote the diameter of a subset $\mathcal{U}$ of $\mathcal{K}(X)$, and let $K$ be a nonempty subset of $X$. Denote

[TABLE]

Let $f_{n}$ be continuous in $X$, $n\geq 0$. Then, system (2.1) induces the following non-autonomous set-valued system:

[TABLE]

where $\bar{f}_{n}$ is defined by $\bar{f}_{n}(A):=f_{n}(A)$ for all $A\in\mathcal{K}(X)$. Thus, $f_{n}(A)\in\mathcal{K}(X)$ and $\bar{f}_{n}$ is a map from $\mathcal{K}(X)$ to $\mathcal{K}(X)$, $n\geq 0$. Denote $\bar{f}_{0,\infty}:=\{\bar{f}_{n}\}_{n=0}^{\infty}$.

Lemma 2.4. ([25], Lemmas II.3.) $\{\bar{f}_{n}\}_{n=0}^{\infty}$* is equi-continuous in $\mathcal{K}(X)$ if and only if $\{f_{n}\}_{n=0}^{\infty}$ is equi-continuous in $X$. Consequently, $\bar{f}:\mathcal{K}(X)\to\mathcal{K}(X)$ is continuous if and only if $f:X\to X$ is continuous.*

The following two lemmas will also be useful in the sequel.

Lemma 2.5 ([10], Lemmas 3.1.1 and 3.1.3).

• (i)

Let $X$ be compact and $K\in\mathcal{K}(X)$. Then, $\langle K\rangle$ is a nonempty compact subset of $\mathcal{K}(X)$, where $\langle K\rangle$ is specified in (2.5).

• (ii)

Let $K_{n}\in\mathcal{K}(X)$, $n\geq 1$. Then, $\langle\bigcap_{n=1}^{\infty}K_{n}\rangle=\bigcap_{n=1}^{\infty}\langle K_{n}\rangle$.

Lemma 2.6. Let $X$ be compact, $f_{n}$ be continuous in $X$, $n\geq 0$, and $K\in\mathcal{K}(X)$. Then, $\langle f_{m}^{-i}(K)\rangle\subset\bar{f}_{m}^{-i}(\langle K\rangle)$ for all $m,i\geq 0$.

Proof. Fix any $m,i\geq 0$. For any $K_{0}\in\langle f_{m}^{-i}(K)\rangle$, one has that $K_{0}\in\mathcal{K}(X)$ and $f_{m}^{i}(K_{0})\subset K$. Since $f_{n}$ is continuous in $X$, $n\geq 0$, one has $f_{m}^{i}(K_{0})\in\mathcal{K}(X)$. Thus, $f_{m}^{i}(K_{0})\in\langle K\rangle$, and so $K_{0}\in\bar{f}_{m}^{-i}(\langle K\rangle)$. Hence, $\langle f_{m}^{-i}(K)\rangle\subset\bar{f}_{m}^{-i}(\langle K\rangle)$. This completes the proof.

2.3. Subshifts of finite type and weak-coupled-expansion for transition matrices

Recall the definitions of subshifts of finite type [38]. A matrix $A=(a_{ij})_{N\times N}$ ($N\geq 2$) is said to be a transition matrix if $a_{ij}=0$ or $1$ for all $i,j$; $\sum_{j=1}^{N}a_{ij}\geq 1$ for all $i$; and $\sum_{i=1}^{N}a_{ij}\geq 1$ for all $j$, $1\leq i,j\leq N$. A transition matrix $A=(a_{ij})_{N\times N}$ is said to be irreducible if, for each pair $1\leq i,j\leq N$, there exists $k\in\mathbf{Z}^{+}$ such that $a^{(k)}_{ij}>0$, where $a^{(k)}_{ij}$ denotes the $(i,j)$ entry of matrix $A^{k}$. For a given transition matrix $A=(a_{ij})_{N\times N}$, denote

[TABLE]

Note that $\Sigma_{N}^{+}(A)$ is a compact metric space with the metric

[TABLE]

where $\hat{d}(a_{i},b_{i})=0$ if $a_{i}=b_{i}$, and $\hat{d}(a_{i},b_{i})=1$ if $a_{i}\neq b_{i}$, $i\geq 0$. The map $\sigma_{A}:\Sigma_{N}^{+}(A)\to\Sigma_{N}^{+}(A)$ with $\sigma_{A}((s_{0},s_{1},s_{2},\cdots)):=(s_{1},s_{2},\cdots)$ is said to be a subshift of finite type associated with $A$. Its topological entropy is equal to $\log\rho(A)$, where

[TABLE]

It is known that $(\Sigma_{N}^{+}(A),\sigma_{A})$ is Li-Yorke chaotic if and only if it is distributionally chaotic [32].

Next, the definition of weak coupled-expansion for a transition matrix is introduced.

Definition 2.4. Let $A=(a_{ij})_{N\times N}$ be a transition matrix. If there exists a sequence $\{V_{i,n}\}_{n=0}^{\infty}$ of nonempty subsets of $X$ with $V_{i,n}\cap V_{j,n}=\partial V_{i,n}\cap\partial V_{j,n}$ ($d(V_{i,n},V_{j,n})>0$) for all $1\leq i\neq j\leq N$ and $n\geq 0$ such that

[TABLE]

then system (2.1) is said to be (strictly) weakly $A$-coupled -expanding in $\{V_{i,n}\}_{n=0}^{\infty}$, $1\leq i\leq N$. In the special case that $V_{i,n}=V_{i}$, $1\leq i\leq N$, $n\geq 0$, it is said to be (strictly) $A$-coupled-expanding in $V_{i}$, $1\leq i\leq N$.

Remark 2.1. Definition 2.4 is a slight revision of that in [35].

Denote

[TABLE]

Then, $U_{i}$, $1\leq i\leq N$, are disjoint and nonempty compact subsets of $\Sigma_{N}^{+}(A)$ and satisfy that

[TABLE]

Lemma 2.7 ([30], Theorem 3.1). $\sigma_{A}$* is strictly $A$-coupled-expanding in $U_{i}$, $1\leq i\leq N$.*

3. Topological (semi)conjugacy and equi-(semi)conjugacy between non-autonomous discrete systems and subshifts of finite type

Now, the topological (semi)conjugacy and equi-(semi)conjugacy between system (2.1) and a subshift of finite type are studied in Sections 3.1 and 3.2, respectively.

3.1. Topological semiconjugacy and conjugacy

Lemma 3.1. Let $A=(a_{ij})_{N\times N}$ be a transition matrix and $\{V_{i,n}\}_{n=0}^{\infty}$ be a sequence of nonempty subsets of $X$, $1\leq i\leq N$.

• (i)

If $f_{n}$ is continuous in $E$, $n\geq 0$, and $V_{i,n}$ is a compact subset of $X$, $1\leq i\leq N$, $n\geq 0$, then $V_{\alpha}^{m,n}$ is a compact subset of $X$ and satisfies that $V_{\alpha}^{m+1,n}\subset V_{\alpha}^{m,n}\subset\bigcup_{i=1}^{N}V_{i,n}$ for all $m,n\geq 0$ and all $\alpha=(a_{0},a_{1},\cdots)\in\Sigma_{N}^{+}(A)$, where

[TABLE]

• (ii)

If system (2.1) is weakly $A$-coupled-expanding in $\{V_{i,n}\}_{n=0}^{\infty}$, $1\leq i\leq N$, then

${\bf(H_{1})}$* $V_{\alpha}^{m,n}\neq\emptyset$ for all $m,n\geq 0$ and all $\alpha\in\Sigma_{N}^{+}(A)$.*

• (iii)

If $V_{i}$, $1\leq i\leq N$, are bounded subsets of $X$ with $V_{i,n}\subset V_{i}$, $n\geq 0$, and there exists $\lambda>1$ such that $d(f_{n}(x),f_{n}(y))\geq\lambda d(x,y)$ for all $x,y\in V_{i,n}$, $1\leq i\leq N$, $n\geq 0$, then

${\bf(H_{2})}$* $d(V_{\alpha}^{m,n})$ uniformly converges to [math] with respect to $n\geq 0$ as $m\to\infty$ for any $\alpha\in\Sigma_{N}^{+}(A)$.*

• (iv)

If assumptions (i)-(iii)* hold, then $\bigcap_{m=0}^{\infty}V_{\alpha}^{m,n}$ is a singleton for any $n\geq 0$ and any $\alpha\in\Sigma_{N}^{+}(A)$.*

Proof. It can be easily verified that assertions (i) and (ii) hold. Assertion (iv) is a direct consequence of assertions (i)-(iii) and Lemma 2.3. Thus, it suffices to show assertion (iii). Fix any $\alpha=(a_{0},a_{1},\cdots)\in\Sigma_{N}^{+}(A)$. For any $m,n\geq 0$ and $x,y\in V_{\alpha}^{m,n}$, $f_{n}^{i}(x),f_{n}^{i}(y)\in V_{a_{i},n+i}$, $0\leq i\leq m$. This, together with the assumption (iii), yields that

[TABLE]

and thus

[TABLE]

Consequently,

[TABLE]

Therefore, $d(V_{\alpha}^{m,n})$ uniformly converges to [math] with respect to $n\geq 0$ as $m\to\infty$. The proof is complete.

Next, a necessary and sufficient condition is derived to ensure system (2.1) to have an invariant subsystem that is topologically semiconjugate to a subshift of finite type.

Theorem 3.1. Let $A=(a_{ij})_{N\times N}$ be a transition matrix. Then, there exists a sequence $\{V_{i,n}\}_{n=0}^{\infty}$ of nonempty compact subsets of $X$ with $V_{i,n}\cap V_{j,n}=\emptyset$ for all $1\leq i\neq j\leq N$ and $n\geq 0$ such that $f_{n}$ is continuous in $E$, $n\geq 0$, and satisfies assumption ${\bf(H_{1})}$, where $E$ is specified in (3.1), if and only if, for any $n\geq 0$, there exists a nonempty compact subset $\Lambda_{n}\subset X$ with $f_{n}(\Lambda_{n})\subset\Lambda_{n+1}$ such that $f_{n}$ is continuous in $\bigcup_{n=0}^{\infty}\Lambda_{n}$ and the invariant subsystem of system (2.1) on $\{\Lambda_{n}\}_{n=0}^{\infty}$ is topologically semiconjugate to $(\Sigma_{N}^{+}(A),\sigma_{A})$.

Proof. Necessity. Fix any $n\geq 0$. Then, $V_{\alpha}^{m,n}$ is a nonempty compact subset of $X$ and satisfies that $V_{\alpha}^{m+1,n}\subset V_{\alpha}^{m,n}$, $m\geq 0$, $\alpha\in\Sigma_{N}^{+}(A)$, by (i) of Lemma 3.1. Thus, $\bigcap_{m=0}^{\infty}V_{\alpha}^{m,n}\neq\emptyset$ for all $\alpha\in\Sigma_{N}^{+}(A)$. Denote

[TABLE]

Clearly, $\Lambda_{n}\neq\emptyset$ and $\Lambda_{n}\subset\bigcup_{i=1}^{N}V_{i,n}$. Then, $\bigcup_{n=0}^{\infty}\Lambda_{n}\subset E$, and thus $f_{n}$ is continuous in $\bigcup_{n=0}^{\infty}\Lambda_{n}$ since $f_{n}$ is continuous in $E$.

Suppose that $\{x_{k}\}_{k=1}^{\infty}\subset\Lambda_{n}$ is a convergent sequence with $x_{k}\to x$ as $k\to\infty$. Then, for any $k\geq 1$, there exists $\alpha_{k}\in\Sigma_{N}^{+}(A)$ such that $x_{k}\in\bigcap_{m=0}^{\infty}V_{\alpha_{k}}^{m,n}$. Since $\Sigma_{N}^{+}(A)$ is compact, without loss of generality, suppose that $\alpha_{k}\to\alpha$ as $k\to\infty$. Then, for any $m\geq 0$, there exists $N_{m}\in\mathbf{Z}^{+}$ such that $V_{\alpha_{k}}^{m,n}=V_{\alpha}^{m,n}$ for all $k\geq N_{m}$. So, $\{x_{k}\}_{k=N_{m}}^{\infty}\subset V_{\alpha}^{m,n}$. Thus, $x\in V_{\alpha}^{m,n}$ as $V_{\alpha}^{m,n}$ is compact. Hence, $x\in\bigcap_{m=0}^{\infty}V_{\alpha}^{m,n}\subset\Lambda_{n}$. Therefore, $\Lambda_{n}$ is closed, and thus compact, since $\bigcup_{i=1}^{N}V_{i,n}$ is compact.

For any $x\in\Lambda_{n}$, there exists $\alpha\in\Sigma_{N}^{+}(A)$ such that $x\in\bigcap_{m=0}^{\infty}V_{\alpha}^{m,n}$. Define $\varphi_{n}(x)=\alpha$. Then, the map $\varphi_{n}:\Lambda_{n}\to\Sigma_{N}^{+}(A)$ is well defined since $V_{i,k}\cap V_{j,k}=\emptyset$ for all $1\leq i\neq j\leq N$ and $k\geq 0$. Clearly, $\varphi_{n}$ is surjective. It is easy to verify that $f_{n}(x)\in\bigcap_{m=0}^{\infty}V_{\sigma_{A}(\alpha)}^{m,n+1}\subset\Lambda_{n+1}$. Thus, $f_{n}(\Lambda_{n})\subset\Lambda_{n+1}$. Moreover, $\varphi_{n+1}\circ f_{n}(x)=\sigma_{A}(\alpha)=\sigma_{A}\circ\varphi_{n}(x)$. Hence, $\varphi_{n+1}\circ f_{n}=\sigma_{A}\circ\varphi_{n}$.

Next, it will be shown that $\varphi_{n}$ is continuous in $\Lambda_{n}$ for any fixed $n\geq 0$. For any $\epsilon>0$, there exists $N_{1}\in\mathbf{Z}^{+}$ such that $2^{-N_{1}}<\epsilon$. Since $\Lambda_{n+j}$ is compact and $f_{n+j}$ is continuous in $\Lambda_{n+j}$, $0\leq j\leq N_{1}-1$, there exists $\delta>0$ such that for any $x,y\in\Lambda_{n}$ with $\varphi_{n}(x)=\alpha=(a_{0},a_{1},\cdots)$ and $\varphi_{n}(y)=\beta=(b_{0},b_{1},\cdots)$, if $d(x,y)<\delta$, then

[TABLE]

This, together with the fact that $f_{n}^{j}(x)\in V_{a_{j},n+j}$ and $f_{n}^{j}(y)\in V_{b_{j},n+j}$, $0\leq j\leq N_{1}$, implies that $a_{j}=b_{j},\;\;0\leq j\leq N_{1}.$ Thus, one has

[TABLE]

Hence, $\varphi_{n}$ is continuous in $\Lambda_{n}$. Therefore, the invariant subsystem of system (2.1) on $\{\Lambda_{n}\}_{n=0}^{\infty}$ is topologically semiconjugate to $(\Sigma_{N}^{+}(A),\sigma_{A})$.

Sufficiency. Suppose that the invariant subsystem of system (2.1) on $\{\Lambda_{n}\}_{n=0}^{\infty}$ is topologically $\{\hat{\varphi}_{n}\}_{n=0}^{\infty}$-semiconjugate to $(\Sigma_{N}^{+}(A),\sigma_{A})$. Then, $\hat{\varphi}_{n}:\Lambda_{n}\to\Sigma_{N}^{+}(A)$ is continuous and surjective, and satisfies that

[TABLE]

Thus,

[TABLE]

Let

[TABLE]

where $U_{i}$ is specified in (2.8). For any $1\leq i\leq N$ and $n\geq 0$, $V_{i,n}\neq\emptyset$ since $\hat{\varphi}_{n}$ is surjective; $V_{i,n}\subset\Lambda_{n}$; and $V_{i,n}$ is compact because $U_{i}$ is closed, $\varphi_{n}$ is continuous, and $\Lambda_{n}$ is compact. It is evident that $E\subset\bigcup_{n=0}^{\infty}\Lambda_{n}$, implying that $f_{n}$ is continuous in $E$, $n\geq 0$. Since $U_{i},\;1\leq i\leq N$, are disjoint, $V_{i,n}\cap V_{j,n}=\emptyset$ for all $1\leq i\neq j\leq N$ and $n\geq 0$. Fix any $m,n\geq 0$ and any $\alpha=(a_{0},a_{1},\cdots)\in\Sigma_{N}^{+}(A)$. It follows from (3.4) and (3.5) that

[TABLE]

By Lemma 2.7, $\bigcap_{k=0}^{m}\sigma_{A}^{-k}(U_{a_{k}})\neq\emptyset$. Hence, $V_{\alpha}^{m,n}\neq\emptyset$, $m,n\geq 0$, $\alpha\in\Sigma_{N}^{+}(A)$. This completes the proof of the theorem.

Now, a sufficient condition is derived to ensure the system (2.1) to have an invariant subsystem that is topologically semiconjugate to a subshift of finite type.

Corollary 3.1. Let assumptions (i)-(ii)* of Lemma 3.1 hold and assume $V_{i,n}\cap V_{j,n}=\emptyset$, $1\leq i\neq j\leq N$, $n\geq 0$. Then, for any $n\geq 0$, there exists a nonempty compact subset $\Lambda_{n}\subset\bigcup_{i=1}^{N}V_{i,n}$ with $f_{n}(\Lambda_{n})=\Lambda_{n+1}$ such that $f_{n}$ is continuous in $\bigcup_{n=0}^{\infty}\Lambda_{n}$ and the invariant subsystem of system (2.1) on $\{\Lambda_{n}\}_{n=0}^{\infty}$ is topologically semiconjugate to $(\Sigma_{N}^{+}(A),\sigma_{A})$.*

Proof. By (ii) of Lemma 3.1 and Theorem 3.1, it suffices to show that $\Lambda_{n+1}\subset f_{n}(\Lambda_{n})$ for all $n\geq 0$, where $\Lambda_{n}$ is specified in (3.2). Fix any $n\geq 0$. For any $y\in\Lambda_{n+1}$, there exists $\beta=(b_{0},b_{1},\cdots)\in\Sigma_{N}^{+}(A)$ such that $y\in\bigcap_{m=0}^{\infty}V_{\beta}^{m,n+1}$ by (3.2). Then, $f_{n+1}^{k}(y)\in V_{b_{k},n+1+k}$ for any $k\geq 0$. Since $\sigma_{A}$ is surjective, there exists $\beta^{\prime}=(a_{0},b_{0},b_{1},\cdots)\in\Sigma_{N}^{+}(A)$ such that $\sigma_{A}(\beta^{\prime})=\beta$. As system (2.1) is weakly $A$-coupled-expanding in $\{V_{i,n}\}_{n=0}^{\infty}$, $V_{b_{0},n+1}\subset f_{n}(V_{a_{0},n})$. Consequently, there exists $x\in V_{a_{0},n}$ such that $y=f_{n}(x)$. So, $f_{n}^{k+1}(x)\in V_{b_{k},n+1+k}$, $k\geq 0$, and thus $x\in\bigcap_{m=0}^{\infty}V_{\beta^{\prime}}^{m,n}\subset\Lambda_{n}$. Hence, $y\in f_{n}(\Lambda_{n})$. Therefore, $\Lambda_{n+1}\subset f_{n}(\Lambda_{n})$ for all $n\geq 0$. The proof is complete.

Next, a necessary and sufficient condition is established under which an invariant subsystem of system (2.1) is topologically conjugate to a subshift of finite type.

Theorem 3.2. Let $A=(a_{ij})_{N\times N}$ be a transition matrix. Then, there exists a sequence $\{V_{i,n}\}_{n=0}^{\infty}$ of nonempty compact subsets of $X$ with $V_{i,n}\cap V_{j,n}=\emptyset$, $1\leq i\neq j\leq N$, $n\geq 0$, such that $f_{n}$ is continuous in $E$, $n\geq 0$, and $\bigcap_{m=0}^{\infty}V_{\alpha}^{m,n}$ is a singleton for any $n\geq 0$ and any $\alpha\in\Sigma_{N}^{+}(A)$ if and only if, for any $n\geq 0$, there exists a nonempty compact subset $\Lambda_{n}\subset X$ with $f_{n}(\Lambda_{n})=\Lambda_{n+1}$ such that $f_{n}$ is continuous in $\bigcup_{n=0}^{\infty}\Lambda_{n}$ and the invariant subsystem of system (2.1) on $\{\Lambda_{n}\}_{n=0}^{\infty}$ is topologically conjugate to $(\Sigma_{N}^{+}(A),\sigma_{A})$.

Proof. Necessity. Fix any $n\geq 0$. Denote

[TABLE]

Clearly, $\Lambda_{n}\neq\emptyset$ and $\Lambda_{n}\subset\bigcup_{i=1}^{N}V_{i,n}$. Then, $\bigcup_{n=0}^{\infty}\Lambda_{n}\subset E$, and thus $f_{n}$ is continuous in $\bigcup_{n=0}^{\infty}\Lambda_{n}$. It can be easily verified that, for any $\alpha\in\Sigma_{N}^{+}(A)$,

[TABLE]

This, together with the fact that $\sigma_{A}$ is surjective, implies that $f_{n}(\Lambda_{n})=\Lambda_{n+1}$.

Define a map $h_{n}:\Sigma_{N}^{+}(A)\to\Lambda_{n}$ by $h_{n}(\alpha)=x^{n}(\alpha)$. Then, $h_{n}$ is well defined, surjective, and one-to-one, since $V_{i,k}\cap V_{j,k}=\emptyset$ for all $1\leq i\neq j\leq N$ and $k\geq 0$. By (3.6), for any $\alpha\in\Sigma_{N}^{+}(A)$, one has

[TABLE]

which yields that

[TABLE]

Next, it will be shown that $h_{n}$ is continuous in $\Sigma_{N}^{+}(A)$ for any fixed $n\geq 0$. Fix any $\alpha\in\Sigma_{N}^{+}(A)$. By the assumption that $\bigcap_{m=0}^{\infty}V_{\alpha}^{m,n}$ is a singleton and using Lemma 2.3, one has that $\lim_{m\to\infty}d(V_{\alpha}^{m,n})=0$. Thus, for any $\epsilon>0$, there exists $N_{2}\in\mathbf{Z}^{+}$ such that $d(V_{\alpha}^{m,n})<\epsilon$ for all $m\geq N_{2}$. Denote $\delta:=2^{-N_{2}}$. For any $\beta\in\Sigma_{N}^{+}(A)$ with $\hat{\rho}(\alpha,\beta)<\delta$, one has that $a_{j}=b_{j},\;\;0\leq j\leq N_{2}$. Then, $x^{n}(\alpha),x^{n}(\beta)\in V_{\alpha}^{N_{2},n}$, and thus

[TABLE]

Therefore, $h_{n}$ is a continuous and bijective map from a compact metric space $(\Sigma_{N}^{+}(A),\hat{\rho})$ to a metric space $(\Lambda_{n},d)$. So, $\Lambda_{n}=h_{n}(\Sigma_{N}^{+}(A))$ is compact and $h_{n}$ is a homeomorphism. Consequently, the invariant subsystem of system (2.1) on $\{\Lambda_{n}\}_{n=0}^{\infty}$ is topologically conjugate to $(\Sigma_{N}^{+}(A),\sigma_{A})$.

Sufficiency. Suppose that the invariant subsystem of system (2.1) on $\{\Lambda_{n}\}_{n=0}^{\infty}$ is topologically $\{\hat{\varphi}_{n}\}_{n=0}^{\infty}$-conjugate to $(\Sigma_{N}^{+}(A),\sigma_{A})$. Then, $\hat{\varphi}_{n}:\Lambda_{n}\to\Sigma_{N}^{+}(A)$ is a homeomorphism, $n\geq 0$, and satisfies (3.3) and (3.4). Let $V_{i,n}$ be specified in (3.5). Then, $\{V_{i,n}\}_{n=0}^{\infty}$ is a sequence of nonempty compact subsets of $X$ with $V_{i,n}\cap V_{j,n}=\emptyset$, $1\leq i\neq j\leq N$, $n\geq 0$, and $f_{n}$ is continuous in $E$, $n\geq 0$. Fix any $n\geq 0$ and any $\alpha=(a_{0},a_{1},\cdots)\in\Sigma_{N}^{+}(A)$. Then, by (3.4) and (3.5), one has that

[TABLE]

Note that $\bigcap_{m=0}^{\infty}\sigma_{A}^{-m}(U_{a_{m}})=\{\alpha\}$ by Proposition 3.1 in [30]. Hence, $\bigcap_{m=0}^{\infty}V_{\alpha}^{m,n}$ is a singleton for any $n\geq 0$ and any $\alpha\in\Sigma_{N}^{+}(A)$. This completes the proof of the theorem.

Remark 3.1. Theorems 3.1 and 3.2 and Corollary 3.1 extend Theorems 4.1, 3.1, and 4.2 in [9] from autonomous discrete systems to non-autonomous discrete systems, respectively.

The following result is a direct consequence of (iv) of Lemma 3.1 and Theorem 3.2.

Corollary 3.2. Let assumptions (i)-(iii)* of Lemma 3.1 hold and suppose that $V_{i,n}\cap V_{j,n}=\emptyset$, $1\leq i\neq j\leq N$, $n\geq 0$. Then, for any $n\geq 0$, there exists a nonempty compact subset $\Lambda_{n}\subset\bigcup_{i=1}^{N}V_{i,n}$ with $f_{n}(\Lambda_{n})=\Lambda_{n+1}$ such that $f_{n}$ is continuous in $\bigcup_{n=0}^{\infty}\Lambda_{n}$ and the invariant subsystem of system (2.1) on $\{\Lambda_{n}\}_{n=0}^{\infty}$ is topologically conjugate to $(\Sigma_{N}^{+}(A),\sigma_{A})$.*

The next result shows that, under certain conditions, topological conjugacy to a subshift of finite type implies weak coupled-expansion for a transition matrix for system (2.1).

Proposition 3.1. Let $A=(a_{ij})_{N\times N}$ be a transition matrix. Assume that, for any $n\geq 0$, there exists a nonempty compact subset $\Lambda_{n}\subset X$ with $f_{n}(\Lambda_{n})=\Lambda_{n+1}$ such that $f_{n}$ is continuous in $\bigcup_{n=0}^{\infty}\Lambda_{n}$ and the invariant subsystem of system (2.1) on $\{\Lambda_{n}\}_{n=0}^{\infty}$ is topologically conjugate to $(\Sigma_{N}^{+}(A),\sigma_{A})$. Then, there exists a sequence $\{V_{i,n}\}_{n=0}^{\infty}$ of nonempty compact subsets of $X$, $1\leq i\leq N$, with $\bigcup_{i=1}^{N}V_{i,n}=\Lambda_{n}$, such that $f_{n}$ is continuous in $E$, $n\geq 0$, and system (2.1) is strictly weakly $A$-coupled-expanding in $\{V_{i,n}\}_{n=0}^{\infty}$, $1\leq i\leq N$.

Proof. Suppose that the invariant subsystem of system (2.1) on $\{\Lambda_{n}\}_{n=0}^{\infty}$ is topologically $\{\hat{\varphi}_{n}\}_{n=0}^{\infty}$-conjugate to $(\Sigma_{N}^{+}(A),\sigma_{A})$. Then, $\hat{\varphi}_{n}:\Lambda_{n}\to\Sigma_{N}^{+}(A)$ is a homeomorphism, $n\geq 0$, and satisfies (3.3). Let $V_{i,n}$ be specified in (3.5). Then, $f_{n}$ is continuous in $E$, $n\geq 0$, and $\{V_{i,n}\}_{n=0}^{\infty}$ is a sequence of nonempty compact subsets of $X$ with $V_{i,n}\cap V_{j,n}=\emptyset$, $1\leq i\neq j\leq N$, $n\geq 0$, since $U_{i},\;1\leq i\leq N$, are disjoint. Thus, $d(V_{i,n},V_{j,n})>0$, $1\leq i\neq j\leq N$, $n\geq 0$. By (2.9) and (3.5), one has that

[TABLE]

Fix any $n\geq 0$ and $1\leq i\leq N$. By (3.3), (3.5), and Lemma 2.7, one has that

[TABLE]

Consequently, system (2.1) is strictly weakly $A$-coupled-expanding in $\{V_{i,n}\}_{n=0}^{\infty}$, $1\leq i\leq N$. The proof is complete.

3.2. Topological equi-semiconjugacy and equi-conjugacy

First, it will be shown that, under certain conditions, system (2.1) has an invariant subsystem that is topologically equi-semiconjugate to a subshift of finite type.

Theorem 3.3. Let $A=(a_{ij})_{N\times N}$ be a transition matrix and $V_{i}$, $1\leq i\leq N$, be nonempty subsets of $X$ with $d(V_{i},V_{j})>0$, $1\leq i\neq j\leq N$. Assume that there exists a nonempty compact subset $V_{i,n}$ of $X$ with $V_{i,n}\subset V_{i}$, $1\leq i\leq N$, $n\geq 0$, such that $\{f_{n}\}_{n=0}^{\infty}$ is equi-continuous in $E$ and satisfies assumption ${\bf(H_{1})}$. Then, for any $n\geq 0$, there exists a nonempty compact subset $\Lambda_{n}\subset\bigcup_{i=1}^{N}V_{i,n}$ with $f_{n}(\Lambda_{n})\subset\Lambda_{n+1}$ such that the invariant subsystem of system (2.1) on $\{\Lambda_{n}\}_{n=0}^{\infty}$ is topologically equi-semiconjugate to $(\Sigma_{N}^{+}(A),\sigma_{A})$. Consequently, $h(f_{0,\infty},\Lambda_{0})$ $\geq\log\rho(A)$ in the case that $(X,d)$ is compact, where $\rho(A)$ is specified in (2.7).

Proof. By Theorem 3.1, it suffices to show that $\{\varphi_{n}\}_{n=0}^{\infty}$ is equi-continuous in $\{\Lambda_{n}\}_{n=0}^{\infty}$, where $\Lambda_{n}$ and $\varphi_{n}:\Lambda_{n}\to\Sigma_{N}^{+}(A)$ are specified in the necessity part of the proof of Theorem 3.1.

For any $\epsilon>0$, there exists $N_{3}\in\mathbf{Z}^{+}$ such that $2^{-N_{3}}<\epsilon$. Since $\{f_{n}\}_{n=0}^{\infty}$ is equi-continuous in $E$, there exists $\delta>0$ such that, for any $n\geq 0$ and any $x,y\in\Lambda_{n}$ with $\varphi_{n}(x)=\alpha=(a_{0},a_{1},\cdots)$ and $\varphi_{n}(y)=\beta=(b_{0},b_{1},\cdots)$, if $d(x,y)<\delta$, then

[TABLE]

This, together with the fact that $f_{n}^{j}(x)\in V_{a_{j}}$ and $f_{n}^{j}(y)\in V_{b_{j}}$, implies that $a_{j}=b_{j}$, $0\leq j\leq N_{3}$. Thus, one has

[TABLE]

Hence, $\{\varphi_{n}\}_{n=0}^{\infty}$ is equi-continuous in $\{\Lambda_{n}\}_{n=0}^{\infty}$. Therefore, the invariant subsystem of system (2.1) on $\{\Lambda_{n}\}_{n=0}^{\infty}$ is topologically equi-semiconjugate to $(\Sigma_{N}^{+}(A),\sigma_{A})$, and consequently $h(f_{0,\infty},\Lambda_{0})$ $\geq h(\sigma_{A},\Sigma_{N}^{+}(A))=\log\rho(A)$ by Lemma 2.2. This completes the proof.

Remark 3.2. The space $X$ is required to be compact in the notion of topological entropy for non-autonomous discrete systems introduced in [13]. So, this condition is needed in the present paper.

Under a stronger and more verifiable condition, the following stronger conclusion can be drawn.

Theorem 3.4. Let all the assumptions of Theorem 3.3 hold, except that assumption ${\bf(H_{1})}$ is replaced by that system (2.1) is weakly $A$-coupled-expanding in $\{V_{i,n}\}_{n=0}^{\infty}$, $1\leq i\leq N$. Then, all the conclusions of Theorem 3.3 hold and $f_{n}(\Lambda_{n})=\Lambda_{n+1}$ for all $n\geq 0$.

Proof. By (ii) of Lemma 3.1, one can verify that all the assumptions of Theorem 3.3 hold. Thus, all the conclusions of Theorem 3.3 can be derived. By the same method used in the proof of Corollary 3.1, one can prove that $f_{n}(\Lambda_{n})=\Lambda_{n+1}$ for all $n\geq 0$. The proof is complete.

Next, some sufficient conditions are derived to ensure a subshift of finite type to be topologically equi-semiconjugate to an invariant subsystem of system (2.1).

Theorem 3.5. Let $A=(a_{ij})_{N\times N}$ be a transition matrix. Assume that there exists a sequence $\{V_{i,n}\}_{n=0}^{\infty}$ of nonempty compact subsets of $X$, $1\leq i\leq N$, such that $f_{n}$ is continuous in $E$, $n\geq 0$, and satisfies assumptions ${\bf(H_{1})}$ and ${\bf(H_{2})}$. Then, for any $n\geq 0$, there exists a nonempty compact subset $\Lambda_{n}\subset\bigcup_{i=1}^{N}V_{i,n}$ with $f_{n}(\Lambda_{n})=\Lambda_{n+1}$ such that $(\Sigma_{N}^{+}(A),\sigma_{A})$ is topologically equi-semiconjugate to the invariant subsystem of system (2.1) on $\{\Lambda_{n}\}_{n=0}^{\infty}$. Consequently, $h(f_{0,\infty},\Lambda_{0})\leq\log\rho(A)$ in the case that $(X,d)$ is compact.

Proof. By Lemma 2.3 and (i) of Lemma 3.1, one has that $\bigcap_{m=0}^{\infty}V_{\alpha}^{m,n}$ is a singleton, $n\geq 0$, $\alpha\in\Sigma_{N}^{+}(A)$. Let $\Lambda_{n}$ and $h_{n}:\Sigma_{N}^{+}(A)\to\Lambda_{n}$, $n\geq 0$, be specified as in the necessity part of the proof of Theorem 3.2. Then, $h_{n}$ is well defined and surjective, and satisfies (3.7) for all $n\geq 0$.

It suffices to show that $\{h_{n}\}_{n=0}^{\infty}$ is equi-continuous in $\Sigma_{N}^{+}(A)$. Fix any $\alpha=(a_{0},a_{1},\cdots)\in\Sigma_{N}^{+}(A)$. It follows from assumption ${\bf(H_{2})}$ that, for any $\epsilon>0$, there exists $N_{4}\in\mathbf{Z}^{+}$ such that $d(V_{\alpha}^{m,n})<\epsilon$, $m\geq N_{4}$, $n\geq 0$. Denote $\delta:=2^{-N_{4}}$. For any $n\geq 0$ and any $\beta=(b_{0},b_{1},\cdots)\in\Sigma_{N}^{+}(A)$ with $\hat{\rho}(\alpha,\beta)<\delta$, one has that $a_{j}=b_{j}$, $0\leq j\leq N_{4}$. So, $h_{n}(\alpha),h_{n}(\beta)\in V_{\alpha}^{N_{4},n}$. Thus

[TABLE]

Hence, $\{h_{n}\}_{n=0}^{\infty}$ is equi-continuous at $\alpha$. This, together with the fact that $(\Sigma_{N}^{+}(A),\hat{\rho})$ is compact, implies that $\{h_{n}\}_{n=0}^{\infty}$ is equi-continuous in $\Sigma_{N}^{+}(A)$ by Lemma 2.5 in [23]. Therefore, $(\Sigma_{N}^{+}(A),\sigma_{A})$ is topologically equi-semiconjugate to the invariant subsystem of system (2.1) on $\{\Lambda_{n}\}_{n=0}^{\infty}$. Consequently, $h(f_{0,\infty},\Lambda_{0})\leq h(\sigma_{A},\Sigma_{N}^{+}(A))=\log\rho(A)$ by Lemma 2.2. This completes the proof.

The following result can be directly derived from Lemma 3.1 and Theorem 3.5.

Corollary 3.3. Let assumptions (i)-(iii)* of Lemma 3.1 hold. Then, all the conclusions of Theorem 3.5 hold.*

Based on Theorems 3.3 and 3.5, it can be shown that, under certain conditions, system (2.1) has an invariant subsystem that is topologically equi-conjugate to a subshift of finite type.

Theorem 3.6. Let all the assumptions of Theorem 3.3 and assumption ${\bf(H_{2})}$ hold. Then, for any $n\geq 0$, there exists a nonempty compact subset $\Lambda_{n}\subset\bigcup_{i=1}^{N}V_{i,n}$ with $f_{n}(\Lambda_{n})=\Lambda_{n+1}$ such that the invariant subsystem of system (2.1) on $\{\Lambda_{n}\}_{n=0}^{\infty}$ is topologically equi-conjugate to $(\Sigma_{N}^{+}(A),\sigma_{A})$. Consequently, $h(f_{0,\infty},\Lambda_{0})=\log\rho(A)$ in the case that $(X,d)$ is compact. Further, if $A$ is irreducible and $\sum_{j=1}^{N}a_{i_{0}j}\geq 2$ for some $1\leq i_{0}\leq N$, then system (2.1) is Li-Yorke $\delta$-chaotic for some $\delta>0$, and is also distributionally chaotic.

Proof. Let $\varphi_{n}:\Lambda_{n}\to\Sigma_{N}^{+}(A)$ and $h_{n}:\Sigma_{N}^{+}(A)\to\Lambda_{n}$, $n\geq 0$, be specified as in the proofs of Theorems 3.3 and 3.5, respectively. Then, $\varphi_{n}=h_{n}^{-1}$, $n\geq 0$, are homeomorphisms, $\{\varphi_{n}\}_{n=0}^{\infty}$ and $\{h_{n}\}_{n=0}^{\infty}$ are equi-continuous in $\{\Lambda_{n}\}_{n=0}^{\infty}$ and $\Sigma_{N}^{+}(A)$, respectively, and (3.7) holds for all $n\geq 0$. Hence, the invariant subsystem of system (2.1) on $\{\Lambda_{n}\}_{n=0}^{\infty}$ is topologically equi-conjugate to $(\Sigma_{N}^{+}(A),\sigma_{A})$, and consequently $h(f_{0,\infty},\Lambda_{0})=\log\rho(A)$ by Lemma 2.2.

If $A$ is irreducible and $\sum_{j=1}^{N}a_{i_{0}j}\geq 2$ for some $1\leq i_{0}\leq N$, then $(\Sigma_{N}^{+}(A),\sigma_{A})$ has an uncountable $\gamma$-scrambled set with $\gamma=1$ by Lemma 4.1 in [29]. This, together with the result of (i) in Lemma 2.1, implies that system (2.1) is Li-Yorke $\delta$-chaotic for some $\delta>0$. It is also distributionally chaotic by (ii) in Lemma 2.1, since $(\Sigma_{N}^{+}(A),\sigma_{A})$ is distributionally chaotic by Theorem 1.4 in [32]. The proof is complete.

One can obtain the following stronger result under some more verifiable conditions.

Theorem 3.7. Let all the assumptions of Theorem 3.4 and assumption $\rm(iii)$ of Lemma 3.1 hold. Then, all the conclusions of Theorem 3.6 hold and system (2.1) is chaotic in the strong sense of Li-Yorke in the case that $A$ is irreducible and $\sum_{j=1}^{N}a_{i_{0}j}\geq 2$ for some $1\leq i_{0}\leq N$.

Proof. It follows from (ii)-(iii) in Lemma 3.1 that all the assumptions of Theorem 3.6 hold, thus all the conclusions of Theorem 3.6 hold. By assumptions of $\rm(iii)$ in Lemma 3.1 that $V_{i}$, $1\leq i\leq N$, are bounded, one has that $\cup_{n=0}^{\infty}\Lambda_{n}$ is bounded, since $\cup_{n=0}^{\infty}\Lambda_{n}\subset\bigcup_{i=1}^{N}V_{i,n}\subset\bigcup_{i=1}^{N}V_{i}$. This implies that system (2.1) is chaotic in the strong sense of Li-Yorke.

By the same method as that used in the proof of Theorem 3.1 in [23], one can get the following result.

Theorem 3.8. Let assumptions (i)-(ii)* of Lemma 3.1 hold, $f_{n}$ be continuous in the compact metric space $(X,d)$, $n\geq 0$, and $V_{i}$, $1\leq i\leq N$, be disjoint nonempty closed subsets of $X$ with $V_{i,n}\subset V_{i}$ for all $n\geq 0$. Then, $h(f_{0,\infty})\geq\log\rho(A)$.*

Remark 3.3.

• (i)

Theorems 3.4, 3.5, 3.7, and 3.8 and Corollary 3.3 generalize Theorems 4.4, 4.1, 4.5, 3.1, and 4.2 in [23], respectively, where only a special case that $V_{i,n}=V_{i}$, $1\leq i\leq N$, $n\geq 0$, is considered.

• (ii)

Theorems 3.4 and 3.7 relax the assumptions of Theorems 4.4 and 4.5 in [23], respectively, since assumption $({\rm ii})_{b}$ of Theorem 4.4 (resp. Theorem 4.5) in [23] is replaced by a weaker one that $\{f_{n}\}_{n=0}^{\infty}$ is equi-continuous in $E$ in Theorems 3.4 and 3.7 here.

• (iii)

Theorem 3.5 and Corollary 3.3 also relax the assumptions of Theorems 4.1 and 4.2 in [23], respectively, since it is only required $f_{n}$ be continuous in $E$, $n\geq 0$, in Theorem 3.5 and Corollary 3.3 here.

• (iv)

Assumption ${\bf(H_{1})}$ in Theorem 3.5 is strictly weaker than assumption (i) of Theorem 4.1 in [23], since the converse of (ii) of Lemma 3.1 is not true in general, even for autonomous dynamical systems (see Example 3.1.1 in [10]).

4. Topological (semi)conjugacy and equi-(semi)conjugacy between induced set-valued systems and subshifts of finite type

First, some relationships of several related dynamical behaviors between system (2.1) and its induced set-valued system (2.6) are established in Section 4.1. Then, by these results, together with the results obtained in Section 3, the topological (semi)conjugacy and equi-(semi)conjugacy between system (2.6) and a subshift of finite type are proved in Sections 4.2 and 4.3, respectively.

4.1. Relationships of some related dynamical behaviors

To start, the following assumption is made.

${\bf(H_{3})}$ Let $(X,d)$ be a compact metric space, $f_{n}$ be continuous in $X$, $n\geq 0$, $\{V_{i,n}\}_{n=0}^{\infty}$ be a sequence of nonempty closed subsets of $X$, $1\leq i\leq N$, and $A$ be a transition matrix.

The following result can be easily verified based on Lemma 2.4 and (i) of Lemma 2.5.

Proposition 4.1. Let assumption ${\bf(H_{3})}$ hold. Then, $\bar{f}_{n}$ is continuous in the compact metric space $(\mathcal{K}(X),\mathcal{H}_{d})$, $n\geq 0$, and $\{\langle V_{i,n}\rangle\}_{n=0}^{\infty}$ is a sequence of nonempty compact subsets of $\mathcal{K}(X)$, $1\leq i\leq N$. In addition, $V_{i,n}\cap V_{j,n}=\emptyset$ if and only if $\langle V_{i,n}\rangle\cap\langle V_{j,n}\rangle=\emptyset$ for all $1\leq i\neq j\leq N$ and $n\geq 0$.

Proposition 4.2. Let assumption ${\bf(H_{3})}$ hold. Fix any $m,n\geq 0$ and any $\alpha=(a_{0},a_{1},\cdots)$ $\in\Sigma_{N}^{+}(A)$. Then, $V_{\alpha}^{m,n}\neq\emptyset$ if and only if $\mathcal{H}_{\alpha}^{m,n}\neq\emptyset$, where

[TABLE]

Proof. Suppose that $V_{\alpha}^{m,n}\neq\emptyset$. It follows from Lemma 2.6 and (ii) of Lemma 2.5 that

[TABLE]

Conversely, suppose that $\mathcal{H}_{\alpha}^{m,n}\neq\emptyset$. Then, there exists $K_{0}\in\mathcal{H}_{\alpha}^{m,n}$. So, $\bar{f}_{n}^{k}(K_{0})=f_{n}^{k}(K_{0})\in\langle V_{a_{k},n+k}\rangle$, and thus $f_{n}^{k}(K_{0})\subset V_{a_{k},n+k}$, $0\leq k\leq m$. Hence,

[TABLE]

Therefore, $V_{\alpha}^{m,n}\neq\emptyset$. The proof is complete.

Proposition 4.3. Let assumption ${\bf(H_{3})}$ hold. Then, $d(V_{\alpha}^{m,n})$ converges to [math] as $m\to\infty$ if and only if $\mathcal{H}_{d}(\mathcal{H}_{\alpha}^{m,n})$ converges to [math] as $m\to\infty$ for any $\alpha=(a_{0},a_{1},\cdots)\in\Sigma_{N}^{+}(A)$ and $n\geq 0$. Further, $d(V_{\alpha}^{m,n})$ uniformly converges to [math] with respect to $n\geq 0$ as $m\to\infty$ if and only if $\mathcal{H}_{d}(\mathcal{H}_{\alpha}^{m,n})$ uniformly converges to [math] with respect to $n\geq 0$ as $m\to\infty$, for any $\alpha\in\Sigma_{N}^{+}(A)$.

Proof. Fix any $\alpha\in\Sigma_{N}^{+}(A)$ and $n\geq 0$. Suppose that $d(V_{\alpha}^{m,n})$ converges to [math] as $m\to\infty$. Then, for any $\epsilon>0$, there exists $M_{1}>0$ such that $d(V_{\alpha}^{m,n})<\epsilon$ for all $m\geq M_{1}$. Fix any $m\geq M_{1}$. For any $K_{1},K_{2}\in\mathcal{H}_{\alpha}^{m,n}$, one has that

[TABLE]

For any $a\in K_{1}$, there exists $b_{1}\in K_{2}$ such that $d(a,K_{2})=d(a,b_{1})$. By (4.3), one has that $f_{n}^{k}(a),f_{n}^{k}(b_{1})\in V_{a_{k},n+k}$ for all $0\leq k\leq m$, hence $a,b_{1}\in V_{\alpha}^{m,n}$. Thus,

[TABLE]

So, $\sup_{a\in K_{1}}d(a,K_{2})\leq\epsilon$. Similarly, one can verify that $\sup_{b\in K_{2}}d(b,K_{1})\leq\epsilon$. This, together with (2.4), implies that $\mathcal{H}_{d}(K_{1},K_{2})\leq\epsilon$. Hence, $\mathcal{H}_{d}(\mathcal{H}_{\alpha}^{m,n})\leq\epsilon$ for all $m\geq M_{1}$. Consequently, $\mathcal{H}_{d}(\mathcal{H}_{\alpha}^{m,n})$ converges to [math] as $m\to\infty$.

Conversely, suppose that $\mathcal{H}_{d}(\mathcal{H}_{\alpha}^{m,n})$ converges to [math] as $m\to\infty$. Then, for any $\epsilon>0$, there exists $M_{2}>0$ such that $\mathcal{H}_{d}(\mathcal{H}_{\alpha}^{m,n})<\epsilon$ for all $m\geq M_{2}$. Fix any $m\geq M_{2}$. For any $x,y\in V_{\alpha}^{m,n}$, one has that $f_{n}^{k}(x),f_{n}^{k}(y)\in V_{a_{k},n+k}$, $0\leq k\leq m$. Thus, $\bar{f}_{n}^{k}(\{x\}),\bar{f}_{n}^{k}(\{y\})\in\langle V_{a_{k},n+k}\rangle$, $0\leq k\leq m$, which implies that $\{x\},\{y\}\in\bigcap_{k=0}^{m}\bar{f}_{n}^{-k}(\langle V_{a_{k},n+k}\rangle)=\mathcal{H}_{\alpha}^{m,n}$. So,

[TABLE]

Hence, $d(V_{\alpha}^{m,n})\leq\epsilon$ for all $m\geq M_{2}$. Consequently, $d(V_{\alpha}^{m,n})$ converges to [math] as $m\to\infty$.

Similarly, one can show that $d(V_{\alpha}^{m,n})$ uniformly converges to [math] with respect to $n\geq 0$ as $m\to\infty$ if and only if $\mathcal{H}_{d}(\mathcal{H}_{\alpha}^{m,n})$ uniformly converges to [math] with respect to $n\geq 0$ as $m\to\infty$, for any $\alpha\in\Sigma_{N}^{+}(A)$. The proof is complete.

Next, it will be shown that the (strictly) weak $A$-coupled-expansion of system (2.1) is equivalent to that of the induced set-valued system (2.6).

Proposition 4.4. Let assumption ${\bf(H_{3})}$ hold. System (2.1) is (strictly) weakly $A$-coupled-expanding in $\{V_{i,n}\}_{n=0}^{\infty}$, $1\leq i\leq N$, if and only if system (2.6) is (strictly) weakly $A$-coupled-expanding in $\{\langle V_{i,n}\rangle\}_{n=0}^{\infty}$, $1\leq i\leq N$.

Proof. Suppose that system (2.1) is weakly $A$-coupled-expanding in $\{V_{i,n}\}_{n=0}^{\infty}$, $1\leq i\leq N$. Fix any $1\leq i\leq N$ and $n\geq 0$. For any $1\leq j\leq N$ with $a_{ij}=1$ and any $K\in\langle V_{j,n+1}\rangle$, one has that $K\in\mathcal{K}(X)$ and $K\subset V_{j,n+1}\subset f_{n}(V_{i,n})$. Let $K_{0}:=f_{n}^{-1}(K)\cap V_{i,n}$. Then, $K_{0}\in\mathcal{K}(X)$, since $f_{n}$ is continuous and both $K$ and $V_{i,n}$ are compact. Thus, $K_{0}\in\langle V_{i,n}\rangle$. It is easy to verify that $f_{n}(K_{0})=K$. Thus, $K\in\bar{f}_{n}(\langle V_{i,n}\rangle)$. Hence, $\langle V_{j,n+1}\rangle\subset\bar{f}_{n}(\langle V_{i,n}\rangle)$. Therefore, system (2.6) is weakly $A$-coupled-expanding in $\{\langle V_{i,n}\rangle\}_{n=0}^{\infty}$, $1\leq i\leq N$.

Conversely, suppose that system (2.6) is weakly $A$-coupled-expanding in $\{\langle V_{i,n}\rangle\}_{n=0}^{\infty}$, $1\leq i\leq N$. Fix any $1\leq i\leq N$ and $n\geq 0$. For any $1\leq j\leq N$ with $a_{ij}=1$ and any $y\in V_{j,n+1}$, one has that $\{y\}\in\langle V_{j,n+1}\rangle\subset\bar{f}_{n}(\langle V_{i,n}\rangle)$. Then, there exists $K_{1}\in\langle V_{i,n}\rangle$ such that $\{y\}=\bar{f}_{n}(K_{1})=f_{n}(K_{1})$. Thus, there exists $x\in K_{1}\subset V_{i,n}$ such that $y=f_{n}(x)\in f_{n}(V_{i,n})$. Hence, $V_{j,n+1}\subset f_{n}(V_{i,n})$. Therefore, system (2.1) is weakly $A$-coupled-expanding in $\{V_{i,n}\}_{n=0}^{\infty}$, $1\leq i\leq N$.

It is easy to verify that $d(V_{i,n},V_{j,n})>0$ if and only if $\mathcal{H}_{d}(\langle V_{i,n}\rangle,\langle V_{j,n}\rangle)>0$, $1\leq i\neq j\leq N$, $n\geq 0$, by Proposition 4.1. Hence, system (2.1) is strictly weakly $A$-coupled-expanding in $\{V_{i,n}\}_{n=0}^{\infty}$, $1\leq i\leq N$, if and only if system (2.6) is strictly weakly $A$-coupled-expanding in $\{\langle V_{i,n}\rangle\}_{n=0}^{\infty}$, $1\leq i\leq N$. This completes the proof.

4.2. Topological semi-conjugacy and conjugacy

To start, the following assumption is made.

${\bf(H_{4})}$ Let assumption ${\bf(H_{3})}$ hold and $V_{i,n}\cap V_{j,n}=\emptyset$ for all $1\leq i\neq j\leq N$ and $n\geq 0$.

First, a sufficient condition is derived for an invariant subsystem of system (2.6) to be topologically semiconjugate to $(\Sigma_{N}^{+}(A),\sigma_{A})$.

Theorem 4.1. Let assumptions ${\bf(H_{1})}$ and ${\bf(H_{4})}$ hold. Then, for any $n\geq 0$, there exists a nonempty compact subset $\mathcal{C}_{n}\subset\bigcup_{i=1}^{N}\langle V_{i,n}\rangle$ with $\bar{f}_{n}(\mathcal{C}_{n})\subset\mathcal{C}_{n+1}$ such that the invariant subsystem of system (2.6) on $\{\mathcal{C}_{n}\}_{n=0}^{\infty}$ is topologically semiconjugate to $(\Sigma_{N}^{+}(A),\sigma_{A})$.

Proof. It follows from Proposition 4.1 that $\bar{f}_{n}$ is continuous in $\mathcal{K}(X)$, $n\geq 0$, and $\{\langle V_{i,n}\rangle\}_{n=0}^{\infty}$ is a sequence of nonempty compact subsets of $\mathcal{K}(X)$, $1\leq i\leq N$, with $\langle V_{i,n}\rangle\cap\langle V_{j,n}\rangle=\emptyset$ for all $1\leq i\neq j\leq N$ and $n\geq 0$. By Proposition 4.2, one has that $\mathcal{H}_{\alpha}^{m,n}\neq\emptyset$ for all $m,n\geq 0$ and any $\alpha\in\Sigma_{N}^{+}(A)$. Hence, all the assumptions of Theorem 3.1 hold for system (2.6). Therefore, the conclusion holds by Theorem 3.1. The proof is complete.

Remark 4.1.

• (i)

By (3.2), (4.2), and (ii) of Lemma 2.5, one has that

[TABLE]

• (ii)

In the special case that $f_{n}=f$ and $V_{i,n}=V_{i}$, $1\leq i\leq N$, $n\geq 0$, Theorem 4.1 is the same as Theorem 3.1.5 in [10] for autonomous discrete systems.

One can obtain the following stronger conclusion under some stronger and more verifiable condition.

Theorem 4.2. Let assumption ${\bf(H_{4})}$ hold. Assume that system (2.1) is weakly $A$-coupled-expanding in $\{V_{i,n}\}_{n=0}^{\infty}$, $1\leq i\leq N$. Then, the conclusion of Theorem 4.1 holds and $\bar{f}_{n}(\mathcal{C}_{n})=\mathcal{C}_{n+1}$ for all $n\geq 0$.

Proof. It follows from Proposition 4.4 that system (2.6) is weakly $A$-coupled-expanding in $\{\langle V_{i,n}\rangle\}_{n=0}^{\infty}$, $1\leq i\leq N$. This, together with the result of Proposition 4.1, implies that all the assumptions of Corollary 3.1 hold for system (2.6). Hence, the conclusions hold by Corollary 3.1. This completes the proof.

Next, it will be shown that, under certain conditions, system (2.6) has an invariant subsystem that is topologically conjugate to $(\Sigma_{N}^{+}(A),\sigma_{A})$.

Theorem 4.3. Let assumptions ${\bf(H_{1})}$ and ${\bf(H_{4})}$ hold. Assume that $d(V_{\alpha}^{m,n})$ converges to [math] as $m\to\infty$ for any $n\geq 0$ and any $\alpha\in\Sigma_{N}^{+}(A)$. Then, for any $n\geq 0$, there exists a nonempty compact subset $\mathcal{C}_{n}\subset\bigcup_{i=1}^{N}\langle V_{i,n}\rangle$ with $\bar{f}_{n}(\mathcal{C}_{n})=\mathcal{C}_{n+1}$ such that the invariant subsystem of system (2.6) on $\{\mathcal{C}_{n}\}_{n=0}^{\infty}$ is topologically conjugate to $(\Sigma_{N}^{+}(A),\sigma_{A})$.

Proof. Fix any $\alpha\in\Sigma_{N}^{+}(A)$ and $n\geq 0$. By (4.1) and Propositions 4.1-4.2, one has that $\mathcal{H}_{\alpha}^{m,n}$ is a nonempty compact subset of $\mathcal{K}(X)$ and satisfies that $\mathcal{H}_{\alpha}^{m+1,n}\subset\mathcal{H}_{\alpha}^{m,n}$ for all $m\geq 0$. It follows from Proposition 4.3 that $\mathcal{H}_{d}(\mathcal{H}_{\alpha}^{m,n})$ converges to [math] as $m\to\infty$. Applying Lemma 2.3, one obtains that $\bigcap_{m=0}^{\infty}\mathcal{H}_{\alpha}^{m,n}$ is a singleton for any fixed $\alpha\in\Sigma_{N}^{+}(A)$ and $n\geq 0$. This, together with the result of Proposition 4.1, implies that all the assumptions of Theorem 3.2 hold for system (2.6). Therefore, the conclusion holds by Theorem 3.2. The proof is complete.

By Lemma 3.1 and Theorem 4.3, one has the following result.

Corollary 4.1. Let assumptions (ii)-(iii)* of Lemma 3.1 and assumption ${\bf(H_{4})}$ hold. Then, the conclusion of Theorem 4.3 holds.*

4.3. Topological equi-semiconjugacy and equi-conjugacy

First, some sufficient conditions are established to ensure the system (2.6) to have an invariant subsystem that is topologically equi-semiconjugate to a subshift of finite type.

Theorem 4.4. Let assumptions ${\bf(H_{1})}$ and ${\bf(H_{3})}$ hold, $\{f_{n}\}_{n=0}^{\infty}$ be equi-continuous in $X$, and $V_{i}$, $1\leq i\leq N$, be disjoint closed subsets of $X$ with $V_{i,n}\subset V_{i}$, $1\leq i\leq N$, $n\geq 0$. Then, for any $n\geq 0$, there exists a nonempty compact subset $\mathcal{C}_{n}\subset\bigcup_{i=1}^{N}\langle V_{i,n}\rangle$ with $\bar{f}_{n}(\mathcal{C}_{n})\subset\mathcal{C}_{n+1}$ such that the invariant subsystem of system (2.6) on $\{\mathcal{C}_{n}\}_{n=0}^{\infty}$ is topologically equi-semiconjugate to $(\Sigma_{N}^{+}(A),\sigma_{A})$. Consequently, $h(\bar{f}_{0,\infty},\mathcal{C}_{0})\geq\log\rho(A)$.

Proof. By (i) of Lemma 2.5, one has that $\langle V_{i}\rangle$ and $\langle V_{i,n}\rangle$, $1\leq i\leq N$, $n\geq 0$, are nonempty compact subsets of $\mathcal{K}(X)$ with $\langle V_{i,n}\rangle\subset\langle V_{i}\rangle$. Since $V_{i}$, $1\leq i\leq N$, are disjoint, $\langle V_{i}\rangle$, $1\leq i\leq N$, are disjoint. Thus, $\mathcal{H}_{d}(\langle V_{i}\rangle,\langle V_{j}\rangle)>0$ for all $1\leq i\neq j\leq N$. It follows from Lemma 2.4 that $\{\bar{f}_{n}\}_{n=0}^{\infty}$ is equi-continuous in $\mathcal{K}(X)$. In addition, $\mathcal{H}_{\alpha}^{m,n}\neq\emptyset$ for all $m,n\geq 0$ and $\alpha\in\Sigma_{N}^{+}(A)$ by Proposition 4.2. Hence, all the assumptions of Theorem 3.3 hold for system (2.6). Therefore, all the conclusions hold by Theorem 3.3. This completes the proof.

Theorem 4.5. Let all the assumptions of Theorem 4.4 hold, except that assumption ${\bf(H_{1})}$ is replaced by that system (2.1) is weakly $A$-coupled-expanding in $\{V_{i,n}\}_{n=0}^{\infty}$, $1\leq i\leq N$. Then, all the conclusions of Theorem 4.4 hold and $\bar{f}_{n}(\mathcal{C}_{n})=\mathcal{C}_{n+1}$ for all $n\geq 0$.

Proof. By the method used in the proof of Theorem 4.4, together with the result of Proposition 4.4, one can show that all the assumptions of Theorem 3.4 hold for system (2.6). Therefore, the conclusions hold by Theorem 3.4. The proof is complete.

Now, a sufficient condition is derived under which $(\Sigma_{N}^{+}(A),\sigma_{A})$ is topologically equi-semiconjugate to an invariant subsystem of system (2.6).

Theorem 4.6. Let assumptions ${\bf(H_{1})}$-${\bf(H_{3})}$ hold. Then, for any $n\geq 0$, there exists a nonempty compact subset $\mathcal{C}_{n}\subset\bigcup_{i=1}^{N}\langle V_{i,n}\rangle$ with $\bar{f}_{n}(\mathcal{C}_{n})=\mathcal{C}_{n+1}$ such that $(\Sigma_{N}^{+}(A),\sigma_{A})$ is topologically equi-semiconjugate to the invariant subsystem of system (2.6) on $\{\mathcal{C}_{n}\}_{n=0}^{\infty}$. Consequently, $h(\bar{f}_{0,\infty},\mathcal{C}_{0})\leq\log\rho(A)$.

Proof. By Propositions 4.1-4.3, all the assumptions of Theorem 3.5 hold for system (2.6). Therefore, the conclusions hold by Theorem 3.5. This completes the proof.

The following result is a direct consequence of Lemma 3.1 and Theorem 4.6.

Corollary 4.2. Let assumptions ${\rm(ii)}$-${\rm(iii)}$ of Lemma 3.1 and assumption ${\bf(H_{3})}$ hold. Then, all the conclusions of Theorem 4.6 hold.

Next, some sufficient conditions are derived to ensure the system (2.6) to have an invariant subsystem that is topologically equi-conjugate to $(\Sigma_{N}^{+}(A),\sigma_{A})$.

Theorem 4.7. Let all the assumptions of Theorem 4.4 and assumption ${\bf(H_{2})}$ hold. Then, for any $n\geq 0$, there exists a nonempty compact subset $\mathcal{C}_{n}\subset\bigcup_{i=1}^{N}\langle V_{i,n}\rangle$ with $\bar{f}_{n}(\mathcal{C}_{n})=\mathcal{C}_{n+1}$ such that the invariant subsystem of system (2.6) on $\{\mathcal{C}_{n}\}_{n=0}^{\infty}$ is topologically equi-conjugate to $(\Sigma_{N}^{+}(A),\sigma_{A})$, and consequently $h(\bar{f}_{0,\infty},\mathcal{C}_{0})=\log\rho(A)$. Further, if $A$ is irreducible and $\sum_{j=1}^{N}a_{i_{0}j}\geq 2$ for some $1\leq i_{0}\leq N$, then system (2.6) is chaotic in the strong sense of Li-Yorke, and is also distributionally chaotic.

Proof. By the method used in the proof of of Theorem 4.4, together with the result of Proposition 4.3, one can verify that all the assumptions of Theorem 3.6 hold for system (2.6). Therefore, all the conclusions of Theorem 3.6 hold. Further, $\cup_{n=0}^{\infty}\mathcal{C}_{n}$ is bounded since $\cup_{n=0}^{\infty}\mathcal{C}_{n}\subset\cup_{i=1}^{N}\langle V_{i}\rangle$. This implies that system (2.6) is chaotic in the strong sense of Li-Yorke, completing the proof.

The following result is a direct consequence of Lemma 3.1 and Theorem 4.7.

Theorem 4.8. Let all the assumptions of Theorem 4.5 and assumption (iii) of Lemma 3.1 hold. Then all the conclusions of Theorem 4.7 hold and system (2.6) is chaotic in the strong sense of Li-Yorke in the case that $A$ is irreducible and $\sum_{j=1}^{N}a_{i_{0}j}\geq 2$ for some $1\leq i_{0}\leq N$.

Proof. It follows from (ii)-(iii) in Lemma 3.1 that all the assumptions of Theorem 4.7 hold, thus all the conclusions of Theorem 4.7 hold. By the assumption that $V_{i}$, $1\leq i\leq N$, are bounded, one has that $\Lambda_{n}$ is bounded, since $\Lambda_{n}\subset\bigcup_{i=1}^{N}V_{i,n}\subset\bigcup_{i=1}^{N}V_{i}$. This implies that system (2.1) is chaotic in the strong sense of Li-Yorke.

Theorem 4.9. Let all the assumptions of Theorem 3.8 hold. Then, $h(\bar{f}_{0,\infty})\geq\log\rho(A)$.

Proof. By (i) of Lemma 2.5, one has that $\langle V_{i}\rangle$, $1\leq i\leq N$, are disjoint nonempty compact subsets of $\mathcal{K}(X)$ with $\langle V_{i,n}\rangle\subset\langle V_{i}\rangle$ for all $1\leq i\leq N$ and $n\geq 0$. Hence, all the assumptions of Theorem 3.8 hold for system (2.6) by Propositions 4.1 and 4.4. Therefore, $h(\bar{f}_{0,\infty})\geq\log\rho(A)$ by Theorem 3.8. This completes the proof.

5. Examples

In this section, two examples are given to illustrate the theoretical results given in Sections 3 and 4.

Example 5.1. Consider system (2.1) with $f_{n}=f_{1}$ or $f_{2}$ for all $n\geq 0$, where

[TABLE]

and

[TABLE]

Then, $\{f_{n}\}_{n=0}^{\infty}$ is a sequence of equi-continuous maps from $[0,3]$ to $[0,3]$. Let

[TABLE]

Then, $V_{1}$ and $V_{2}$ are nonempty, disjoint, and compact subsets of $[0,3]$. Denote

[TABLE]

In the case that $n\in\mathbf{N}^{1}$, set

[TABLE]

Then, $V_{1,n}\subset V_{1}$, $V_{2,n}\subset V_{2}$, and $f_{n}(V_{1,n})=f_{n}(V_{2,n})=[0,3]$. Thus,

[TABLE]

In the case that $n\in\mathbf{N}\setminus\mathbf{N}^{1}$, set

[TABLE]

Then, $V_{1,n}\subset V_{1}$, $V_{2,n}\subset V_{2}$, and $f_{n}(V_{1,n})=f_{n}(V_{2,n})=[0,3]$. Thus,

[TABLE]

It follows from (5.1) and (5.2) that system (2.1) is strictly weakly $A$-coupled expanding in $\{V_{i,n}\}_{n=0}^{\infty}$, $i=1,2$, with $A=(a_{ij})_{2\times 2}$ and $a_{ij}=1$, $i,j=1,2$. It is evident that $A$ is irreducible and $\sum_{j=1}^{2}a_{i_{0}j}=2$, $i_{0}=1,2$. On the other hand, one can easily verify that

[TABLE]

and

[TABLE]

Hence, all the assumptions of Theorems 3.7, 3.8, and 4.8 hold for system (2.1).

By Theorem 3.7, for any $n\geq 0$, there exists a nonempty compact subset $\Lambda_{n}\subset\bigcup_{i=1}^{2}V_{i,n}\subset\bigcup_{i=1}^{2}V_{i}$ with $f_{n}(\Lambda_{n})=\Lambda_{n+1}$ such that the invariant subsystem of system (2.1) on $\{\Lambda_{n}\}_{n=0}^{\infty}$ is topologically equi-conjugate to $(\Sigma_{2}^{+}(A),\sigma_{A})$, and consequently $h(f_{0,\infty},\Lambda_{0})=\log\rho(A)=\log 2$. Moreover, system (2.1) is chaotic in the strong sense of Li-Yorke, and is also distributionally chaotic.

By Theorem 3.8, one has that $h(f_{0,\infty})\geq\log\rho(A)=\log 2$.

By Theorem 4.8, for any $n\geq 0$, there exists a nonempty compact subset $\mathcal{C}_{n}\subset\bigcup_{i=1}^{2}\langle V_{i,n}\rangle$ $\subset\bigcup_{i=1}^{2}\langle V_{i}\rangle$ with $\bar{f}_{n}(\mathcal{C}_{n})=\mathcal{C}_{n+1}$ such that the invariant subsystem of system (2.6) on $\{\mathcal{C}_{n}\}_{n=0}^{\infty}$ is topologically equi-conjugate to $(\Sigma_{2}^{+}(A),\sigma_{A})$, and consequently $h(\bar{f}_{0,\infty},\mathcal{C}_{0})=\log\rho(A)=\log 2$. Moreover, system (2.6) is chaotic in the strong sense of Li-Yorke, and is also distributionally chaotic.

Example 5.2. Consider the following planar non-autonomous discrete system:

[TABLE]

where $(x_{1},x_{2})\in\mathbf{R}^{2}$ with

[TABLE]

[TABLE]

with $saw_{2}(t)$ being the sawtooth function defined by

[TABLE]

It is evident that $\mathbf{F}_{n}$ is continuous in $\mathbf{R}^{2}$ for all $n\geq 0$. Set

[TABLE]

Clearly, $V_{1}$ and $V_{2}$ are nonempty, disjoint, and compact sets with $d(V_{1},V_{2})>0$.

Next, it is to show that $V_{1}\cup V_{2}\subset\mathbf{F}_{n}(V_{1})\cap\mathbf{F}_{n}(V_{2})$ for any fixed $n\geq 0$. For any $(x_{1},x_{2})\in V_{1}$, one has that

[TABLE]

It follows from (5.4) that, for any $(x_{1},x_{2})\in V_{1}$ with $x_{1}=-1/6$,

[TABLE]

and, for any $(x_{1},x_{2})\in V_{1}$ with $x_{1}=1/6$,

[TABLE]

It then follows from (5.4) that, for any $(x_{1},x_{2})\in V_{1}$ with $x_{2}=-1/6$,

[TABLE]

and, for any $(x_{1},x_{2})\in V_{1}$ with $x_{2}=1/6$,

[TABLE]

Since $\mathbf{F}_{n}$ is continuous in $V_{1}$, by the intermediate value theorem and (5.5)-(5.8), one has that $V_{1}\cup V_{2}\subset\mathbf{F}_{n}(V_{1})$. On the other hand, for any $(x_{1},x_{2})\in V_{2}$, one has that

[TABLE]

With a similar argument, one can show that $V_{1}\cup V_{2}\subset\mathbf{F}_{n}(V_{2})$. Hence, system (5.3) is strictly $A$-coupled-expanding in $V_{1}$ and $V_{2}$ with $A=(a_{ij})_{2\times 2}$ and $a_{ij}=1$, $i,j=1,2$.

By (5.4) and (5.9), one can easily verify that, for any $n\geq 0$,

[TABLE]

where

[TABLE]

Hence, $\{\mathbf{F}_{n}\}_{n=0}^{\infty}$ is equi-continuous in $\cup_{i=1}^{2}{V_{i}}$. Thus, all the assumptions of Theorem 3.7 hold for system (5.3). By Theorem 3.7, one has that, for any $n\geq 0$, there exists a nonempty compact subset $\Lambda_{n}\subset\bigcup_{i=1}^{2}V_{i}$ with $\mathbf{F}_{n}(\Lambda_{n})=\Lambda_{n+1}$ such that the invariant subsystem of system (5.3) on $\{\Lambda_{n}\}_{n=0}^{\infty}$ is topologically equi-conjugate to $(\Sigma_{2}^{+}(A),\sigma_{A})$, and consequently system (5.3) is chaotic in the strong sense of Li-Yorke and is also distributionally chaotic.

Similarly, all the assumptions of Theorem 4.8 hold for the invariant subsystem of system (5.3) on $\{\Lambda_{n}\}_{n=0}^{\infty}$. It follows from Theorem 4.8 that the induced set-valued system of invariant subsystem of system (5.3) on $\{\Lambda_{n}\}_{n=0}^{\infty}$ has an invariant subsystem that is topologically equi-conjugate to $(\Sigma_{2}^{+}(A),\sigma_{A})$, and consequently the induced set-valued system of system (5.3) is chaotic in the strong sense of Li-Yorke and is also distributionally chaotic.

It is remarked here that Theorem 4.8 still holds true when $f_{n}:X\to X$ is replaced by that $f_{n}:\Lambda_{n}\to\Lambda_{n+1}$ with compact subsets $\Lambda_{n}\subset X$ for all $n\geq 0$. Here, $\bigcup_{i=1}^{2}V_{i}$ can be regarded as the base compact space $X$.

Remark 5.1. Some simulation results are displayed in Figures 1 and 2, which show the complicated dynamical behaviors of system (5.3) in Example 5.2 and its induced set-valued system. It is noted that the constructions of these two examples are motivated by [8, 29, 35].

Acknowledgments

This research was supported by the Hong Kong Research Grants Council (GRF Grant CityU11200317) and the NNSF of China (Grant 11571202). The authors would like to thank Dr. Yang Lou for his assistance in simulation.