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- Open Access

# Weakly mixing sets and transitive sets for non-autonomous discrete systems

- Lei Liu
^{1, 2}Email author and - Yuejuan Sun
^{1}

**2014**:217

https://doi.org/10.1186/1687-1847-2014-217

© Liu and Sun; licensee Springer. 2014

**Received:**15 May 2014**Accepted:**15 July 2014**Published:**4 August 2014

## Abstract

In this paper we mainly study the weakly mixing sets and transitive sets of non-autonomous discrete systems. Some basic concepts are introduced for non-autonomous discrete systems, including a weakly mixing set and a transitive set. We discuss the basic properties of weakly mixing sets and transitive sets of non-autonomous discrete systems. Also, we investigate the relationship between two conjugated non-autonomous discrete systems on weakly mixing sets and transitive sets.

**MSC:**54H20, 37B20.

## Keywords

- non-autonomous discrete system
- weakly mixing set
- transitive set

## 1 Introduction

*X*be a topological space, let ${f}_{n}:X\to X$ for each $n\in \mathbb{N}$ be a continuous map, and let ${f}_{1,\mathrm{\infty}}$ denote the sequence $({f}_{1},{f}_{2},\dots ,{f}_{n},\dots )$. The pair $(X,{f}_{1,\mathrm{\infty}})$ is referred to as a non-autonomous discrete system [1]. Define

*X*. In particular, when ${f}_{1,\mathrm{\infty}}$ is a constant sequence $(f,f,\dots ,f,\dots )$, the pair $(X,{f}_{1,\mathrm{\infty}})$ is just a classical discrete dynamical system (autonomous discrete dynamical system) $(X,f)$. The orbit initiated from $x\in X$ under ${f}_{1,\mathrm{\infty}}$ is defined by the set

Its long-term behaviors are determined by its limit sets.

Topological transitivity, weak mixing and sensitive dependence on initial conditions (see [1–4]) are global characteristics of topological dynamical systems. Let $(X,f)$ be a topological dynamical system. $(X,f)$ is *topologically transitive* if for any nonempty open subsets *U* and *V* of *X* there exists $n\in \mathbb{N}$ such that ${f}^{n}(U)\cap V\ne \mathrm{\varnothing}$. $(X,f)$ is (*topologically*) *mixing* if for any nonempty open subsets *U* and *V* of *X*, there exists $N\in \mathbb{N}$ such that ${f}^{n}(U)\cap V\ne \mathrm{\varnothing}$ for all $n\in \mathbb{N}$ with $n\ge N$. $(X,f)$ is (*topologically*) *weakly mixing* if for any nonempty open subsets ${U}_{1}$, ${U}_{2}$, ${V}_{1}$ and ${V}_{2}$ of *X*, there exists $n\in \mathbb{N}$ such that ${f}^{n}({U}_{1})\cap {V}_{1}\ne \mathrm{\varnothing}$ and ${f}^{n}({U}_{2})\cap {V}_{2}\ne \mathrm{\varnothing}$. It follows from these definitions that mixing implies weak mixing which in turn implies transitivity.

Blanchard introduced overall properties and partial properties in [5]. For example, sensitive dependence on initial conditions, Devaney chaos (see [6]), weak mixing, mixing and more belong to overall properties; Li-Yorke chaos (see [7]) and positive entropy (see [2, 8]) belong to partial properties. Weak mixing is an overall property, it is stable under semi-conjugate maps and implies Li-Yorke chaos. By [9], we know that a weakly mixing system always contains a dense uncountable scrambled set. In [10], Blanchard and Huang introduced the concepts of weakly mixing set and partial weak mixing, derived from a result given by Xiong and Yang [11] and showed that ‘partial weak mixing implies Li-Yorke chaos’ and ‘Li-Yorke chaos cannot imply partial weak mixing’. Let *A* be a closed subset of *X* but not a singleton. Then *A* is a *weakly mixing set* of *X* if and only if for any $k\in \mathbb{N}$, any choice of nonempty open subsets ${V}_{1},{V}_{2},\dots ,{V}_{k}$ of *A* and nonempty open subsets ${U}_{1},{U}_{2},\dots ,{U}_{k}$ of *X* with $A\cap {U}_{i}\ne \mathrm{\varnothing}$, $i=1,2,\dots ,k$, there exists $m\in \mathbb{N}$ such that ${f}^{m}({V}_{i})\cap {U}_{i}\ne \mathrm{\varnothing}$ for $1\le i\le k$. $(X,f)$ is called *partial weak mixing* if *X* contains a weakly mixing subset. Next, Oprocha and Zhang [12] extended the notion of weakly mixing set and gave the concept of ‘transitive set’ and discussed its basic properties. Let *A* be a nonempty subset of *X*. *A* is called a *transitive set* of $(X,f)$ if for any choice of a nonempty open subset ${V}^{A}$ of *A* and a nonempty open subset *U* of *X* with $A\cap U\ne \mathrm{\varnothing}$, there exists $n\in \mathbb{N}$ such that ${f}^{n}({V}^{A})\cap U\ne \mathrm{\varnothing}$.

In past ten years, a large number of papers have been devoted to dynamical properties in non-autonomous discrete systems. Kolyada and Snoha [1] gave the definition of topological entropy in non-autonomous discrete systems; Kolyada *et al.* [13] discussed minimality of non-autonomous discrete systems; Kempf [14] and Canovas [15] studied *ω*-limit sets in non-autonomous discrete systems. Krabs [16] discussed stability in non-autonomous discrete systems; Huang *et al.* [17, 18] studied topological pressure and pre-image entropy of non-autonomous discrete systems. Shi and Chen [19] and Oprocha and Wilczynski [20] and Canovas [21] discussed chaos in non-autonomous discrete systems, respectively. Kuang and Cheng [22] studied fractal entropy of non-autonomous systems. In this paper, we extend the notions of weakly mixing set and transitive set and give the definitions of transitive set and weakly mixing set for a non-autonomous discrete system. We discuss the basic properties of weakly mixing sets and transitive sets for non-autonomous discrete systems. Moreover, we investigate the weakly mixing sets and transitive sets for the conjugated non-autonomous discrete systems and obtain that if a system has a transitive set (a weakly mixing set), then the conjugated system has a transitive set (a weakly mixing set).

## 2 Preliminaries

*A*, respectively. ${f}_{1}^{n}$ denotes ${f}_{n}\circ {f}_{n-1}\circ \cdots \circ {f}_{2}\circ {f}_{1}$,

*i.e.*, ${f}_{1}^{n}={f}_{n}\circ {f}_{n-1}\circ \cdots \circ {f}_{2}\circ {f}_{1}$ for any $n\in \mathbb{N}$. We define

for any $k,n\in \mathbb{N}$.

A non-autonomous discrete dynamical system $(X,{f}_{1,\mathrm{\infty}})$ is said to be point transitive if there exists a point $x\in X$, the orbit of *x* is dense in *X*, *i.e.*, $\overline{\gamma (x,{f}_{1,\mathrm{\infty}})}=X$, and *x* is called a transitive point of $(X,{f}_{1,\mathrm{\infty}})$. $(X,{f}_{1,\mathrm{\infty}})$ is said to be topologically transitive if for any two nonempty open sets *U* and *V* of *X*, there exists $k\in \mathbb{N}$ such that ${f}_{1}^{k}(U)\cap V\ne \mathrm{\varnothing}$. $(X,{f}_{1,\mathrm{\infty}})$ is said to be weakly mixing if for any nonempty open sets ${U}_{i}$ and ${V}_{i}$ of *X* for $i=1,2$, there exists $k\in \mathbb{N}$ such that ${f}_{1}^{k}({U}_{i})\cap {V}_{i}\ne \mathrm{\varnothing}$ for $i=1,2$.

**Definition 2.1** [13]

Let $(X,{f}_{1,\mathrm{\infty}})$ be a non-autonomous discrete system. The set $A\subseteq X$ is said to be invariant if ${f}_{1}^{n}(A)\subseteq A$ for any $n\in \mathbb{N}$.

**Definition 2.2** Let $(X,{f}_{1,\mathrm{\infty}})$ be a non-autonomous discrete system and *A* be a nonempty closed subset of *X*. *A* is called a transitive set of $(X,{f}_{1,\mathrm{\infty}})$ if for any choice of a nonempty open set ${V}^{A}$ of *A* and a nonempty open set *U* of *X* with $A\cap U\ne \mathrm{\varnothing}$, there exists $n\in \mathbb{N}$ such that ${f}_{1}^{n}({V}^{A})\cap U\ne \mathrm{\varnothing}$.

**Remark** If $(X,{f}_{1,\mathrm{\infty}})$ is topologically transitive, then *X* is a transitive set of $(X,{f}_{1,\mathrm{\infty}})$.

**Definition 2.3** Let $(X,{f}_{1,\mathrm{\infty}})$ be a non-autonomous discrete system and *A* be a nonempty closed subset of *X* but not a singleton. *A* is called a weakly mixing set of $(X,{f}_{1,\mathrm{\infty}})$ if for any $k\in \mathbb{N}$, any choice of nonempty open subsets ${V}_{1}^{A},{V}_{1}^{A},\dots ,{V}_{k}^{A}$ of *A* and nonempty open subsets ${U}_{1},{U}_{2},\dots ,{U}_{k}$ of *X* with $A\cap {U}_{i}\ne \mathrm{\varnothing}$, $i=1,2,\dots ,k$, there exists $n\in \mathbb{N}$ such that ${f}_{1}^{n}({V}_{i}^{A})\cap {U}_{i}\ne \mathrm{\varnothing}$ for each $1\le i\le k$.

According to the definitions of transitive set and weakly mixing set of a non-autonomous discrete system, we have the following results.

Result 1. If *A* is a weakly mixing set of $(X,{f}_{1,\mathrm{\infty}})$, then *A* is a transitive set of $(X,{f}_{1,\mathrm{\infty}})$.

Result 2. If $a\in X$ is a transitive point of $(X,{f}_{1,\mathrm{\infty}})$, then $\{a\}$ is a transitive set of $(X,{f}_{1,\mathrm{\infty}})$.

**Example 2.1**Let

and ${f}_{1}={f}_{2}=\mathrm{id}$, the identity on $[0,1]$.

which is known to be topologically transitive on $I=[0,1]$ from [6, 8]. We can easily prove that $[0,\frac{1}{2}]$ is a transitive set of $(X,{f}_{1,\mathrm{\infty}})$.

*f*and the 2nd iterate ${f}^{2}$ of the tent map

*f*, respectively.

**Definition 2.4** [23]

Let $(X,\tau )$ be a topological space and *A* be a nonempty set of *X*. *A* is a regular closed set of *X* if $A=\overline{int(A)}$.

We easily prove that *A* is a regular closed set if and only if $int({V}^{A})\ne \mathrm{\varnothing}$ for any nonempty set ${V}^{A}$ of *A*.

**Definition 2.5** [24]

Let $(X,\tau )$ be a topological space. *A* and *B* are two nonempty subsets of *X*. *B* is dense in *A* if $A\subseteq \overline{A\cap B}$.

In fact, we easily prove that *B* is dense in *A* if and only if ${V}^{A}\cap B\ne \mathrm{\varnothing}$ for any nonempty open set ${V}^{A}$ of *A*.

## 3 Main results

In this section, we discuss the properties of transitive sets and weakly mixing sets for non-autonomous discrete systems.

**Proposition 3.1**

*Let*$(X,{f}_{1,\mathrm{\infty}})$

*be a non*-

*autonomous discrete system and*

*A*

*be a nonempty closed set of*

*X*.

*Then the following conditions are equivalent*.

- (1)
*A**is a transitive set of*$(X,{f}_{1,\mathrm{\infty}})$. - (2)
*Let*${V}^{A}$*be a nonempty open subset of**A**and**U**be a nonempty open subset of**X**with*$A\cap U\ne \mathrm{\varnothing}$.*Then there exists*$n\in \mathbb{N}$*such that*${V}^{A}\cap {({f}_{1}^{n})}^{-1}(U)\ne \mathrm{\varnothing}$. - (3)
*Let**U**be a nonempty open set of**X**with*$U\cap A\ne \mathrm{\varnothing}$.*Then*${\bigcup}_{n\in \mathbb{N}}{({f}_{1}^{n})}^{-1}(U)$*is dense in**A*.

*Proof*(1) ⇒ (2) Let

*A*be a transitive set of $(X,{f}_{1,\mathrm{\infty}})$. Then, for any choice of a nonempty open set ${V}^{A}$ of

*A*and a nonempty open set

*U*of

*X*with $A\cap U\ne \mathrm{\varnothing}$, there exists $n\in \mathbb{N}$ such that ${f}_{1}^{n}({V}^{A})\cap U\ne \mathrm{\varnothing}$. Since ${f}_{1}^{n}({V}^{A}\cap {({f}_{1}^{n})}^{-1}(U))={f}_{1}^{n}({V}^{A})\cap U$, it follows that ${V}^{A}\cap {({f}_{1}^{n})}^{-1}(U)\ne \mathrm{\varnothing}$.

- (2)⇒ (3) Let ${V}^{A}$ be any nonempty open set of
*A*and*U*be a nonempty open set of*X*with $A\cap U\ne \mathrm{\varnothing}$. By the assumption of (2), there exists $n\in \mathbb{N}$ such that ${V}^{A}\cap {({f}_{1}^{n})}^{-1}U\ne \mathrm{\varnothing}$. Furthermore, we have${V}^{A}\cap \bigcup _{n\in \mathbb{N}}{\left({f}_{1}^{n}\right)}^{-1}U=\bigcup _{n\in \mathbb{N}}({V}^{A}\cap {\left({f}_{1}^{n}\right)}^{-1}(U))\ne \mathrm{\varnothing}.$

*A*.

- (3)
⇒ (1) Let ${V}^{A}$ be any nonempty open set of

*A*and*U*be a nonempty open set of*X*with $A\cap U\ne \mathrm{\varnothing}$. Since ${\bigcup}_{n\in \mathbb{N}}{({f}_{1}^{n})}^{-1}(U)$ is dense in*A*, it follows that ${V}^{A}\cap {\bigcup}_{n\in \mathbb{N}}{({f}_{1}^{n})}^{-1}(U)\ne \mathrm{\varnothing}$. Furthermore, there exists $n\in \mathbb{N}$ such that ${V}^{A}\cap {({f}_{1}^{n})}^{-1}(U)\ne \mathrm{\varnothing}$. As ${f}_{1}^{n}({V}^{A}\cap {({f}_{1}^{n})}^{-1}(U))={f}_{1}^{n}({V}^{A})\cap U$, we have ${f}_{1}^{n}({V}^{A})\cap U\ne \mathrm{\varnothing}$. Therefore,*A*is a transitive set of $(X,{f}_{1,\mathrm{\infty}})$. □

**Corollary 3.1**

*Let*$(X,f)$

*be a classical dynamical system and*

*A*

*be a nonempty closed set of*

*X*.

*Then the following conditions are equivalent*.

- (1)
*A**is a transitive set of*$(X,f)$. - (2)
*Let*${V}^{A}$*be a nonempty open subset of**A**and**U**be a nonempty open subset of**X**with*$A\cap U\ne \mathrm{\varnothing}$.*Then there exists*$n\in \mathbb{N}$*such that*${V}^{A}\cap {f}^{-n}(U)\ne \mathrm{\varnothing}$. - (3)
*Let**U**be a nonempty open set of**X**with*$A\cap U\ne \mathrm{\varnothing}$.*Then*${\bigcup}_{n\in \mathbb{N}}{f}^{-n}(U)$*is dense in**A*.

**Proposition 3.2**

*Let*$(X,{f}_{1,\mathrm{\infty}})$

*be a non*-

*autonomous discrete system*,

*where*$(X,d)$

*is a metric space and*

*A*

*is a nonempty closed subset of*

*X*.

*Then the following conditions are equivalent*.

- (1)
*A**is a transitive set of*$(X,{f}_{1,\mathrm{\infty}})$. - (2)
*Let*$a,x\in A$*and*$\epsilon ,\delta >0$.*Then there exists*$n\in \mathbb{N}$*such that*$(A\cap B(a,\epsilon ))\cap {({f}_{1}^{n})}^{-1}(B(x,\epsilon ))\ne \mathrm{\varnothing}$. - (3)
*Let*$a,x\in A$*and*$\epsilon >0$.*Then there exists*$n\in \mathbb{N}$*such that*$(A\cap B(a,\epsilon ))\cap {({f}_{1}^{n})}^{-1}(B(x,\epsilon ))\ne \mathrm{\varnothing}$.

*Proof*(1) ⇒ (2) By the definition of transitive set, (2) is obtained easily.

- (2)
⇒ (3) Obviously.

- (3)
⇒ (1) Let ${V}^{A}$ be any nonempty open set of

*A*and*U*be a nonempty open set of*X*with $A\cap U\ne \mathrm{\varnothing}$, then there exist $a,x\in A$ and $\epsilon >0$ such that $A\cap B(a,\epsilon )\subseteq {V}^{A}$ and $B(x,\epsilon )\subseteq U$. By the assumption of (3), there exists $n\in \mathbb{N}$ such that $(A\cap B(a,\epsilon ))\cap {({f}_{1}^{n})}^{-1}(B(x,\epsilon ))\ne \mathrm{\varnothing}$, further, ${V}^{A}\cap {({f}_{1}^{n})}^{-1}(U)\ne \mathrm{\varnothing}$. Therefore,*A*is a transitive set of $(X,{f}_{1,\mathrm{\infty}})$. □

**Proposition 3.3**

*Let*$(X,{f}_{1,\mathrm{\infty}})$

*be a non*-

*autonomous discrete system and*

*A*

*is a transitive set of*$(X,{f}_{1,\mathrm{\infty}})$.

*Then*:

- (1)
*U**is dense in**A**if**U**is a nonempty open set of**X**satisfying*$A\cap U\ne \mathrm{\varnothing}$*and*${({f}_{1}^{n})}^{-1}(U)\subseteq U$*for every*$n\in \mathbb{N}$. - (2)
$E=A$

*or**E**is nowhere dense in**A**if**E**is a closed invariant subset of**X**and*$E\subseteq A$. - (3)
${\bigcup}_{n\in \mathbb{N}}{f}_{1}^{n}(A)$

*is dense in**A**if**A**is a regular closed set of**X*.

*Proof*(1) Since ${({f}_{1}^{n})}^{-1}(U)\subseteq U$ for every $n\in \mathbb{N}$, we have ${\bigcup}_{n\in \mathbb{N}}{({f}_{1}^{n})}^{-1}(U)\subseteq U$. By Proposition 3.1, we have that

*U*is dense in

*A*.

- (2)Let $E\ne A$. Since
*E*is a closed set of*X*and $E\subseteq A$, it follows that $U=X\setminus E$ is an open set of*X*and $U\cap A\ne \mathrm{\varnothing}$. Moreover,*E*is an invariant subset of*X*, we have ${f}_{1}^{n}(E)\subseteq E$ for every $n\in \mathbb{N}$. Furthermore,${\left({f}_{1}^{n}\right)}^{-1}(U)={\left({f}_{1}^{n}\right)}^{-1}(X\setminus E)={\left({f}_{1}^{n}\right)}^{-1}(X)\setminus {\left({f}_{1}^{n}\right)}^{-1}(E)\subseteq X\setminus E=U\phantom{\rule{1em}{0ex}}\text{for every}n\in \mathbb{N}.$

*U*is dense in

*A*. Therefore,

*E*is nowhere dense in

*A*.

- (3)
Let ${V}^{A}$ be a nonempty open set of

*A*. Since*A*is a regular closed set of*X*, it follows that $int({V}^{A})\ne \mathrm{\varnothing}$ and $int(A)\ne \mathrm{\varnothing}$. Moreover,*A*is a transitive set of $(X,{f}_{1,\mathrm{\infty}})$, there exists $n\in \mathbb{N}$ such that ${f}_{1}^{n}(int(A))\cap int({V}^{A})\ne \mathrm{\varnothing}$. Furthermore, we have ${f}_{1}^{n}(A)\cap {V}^{A}\ne \mathrm{\varnothing}$, which implies that ${\bigcup}_{n\in \mathbb{N}}{f}_{1}^{n}(A)$ is dense in*A*. □

**Theorem 3.1** *Let* $(X,{f}_{1,\mathrm{\infty}})$ *be a non*-*autonomous discrete system and* *A* *be a nonempty closed invariant set of* *X*. *Then* *A* *is a transitive set of* $(X,{f}_{1,\mathrm{\infty}})$ *if and only if* $(A,{f}_{1,\mathrm{\infty}})$ *is topologically transitive*.

*Proof* Necessity. Let ${V}^{A}$ and ${U}^{A}$ be two nonempty open subsets of *A*. For a nonempty open subset ${U}^{A}$ of *A*, there exists an open set *U* of *X* such that ${U}^{A}=U\cap A$. Since *A* is a transitive set of $(X,{f}_{1,\mathrm{\infty}})$, there exists $n\in \mathbb{N}$ such that ${f}_{1}^{n}({V}^{A})\cap U\ne \mathrm{\varnothing}$. Moreover, *A* is invariant, *i.e.*, ${f}_{1}^{n}(A)\subseteq A$ for every $n\in \mathbb{N}$, which implies that ${f}_{1}^{n}({V}^{A})\subseteq A$. Therefore, ${f}_{1}^{n}({V}^{A})\cap A\cap U\ne \mathrm{\varnothing}$, *i.e.*, ${f}_{1}^{n}({V}^{A})\cap {U}^{A}\ne \mathrm{\varnothing}$. This shows that $(A,{f}_{1,\mathrm{\infty}})$ is topologically transitive.

Sufficiency. Let ${V}^{A}$ be a nonempty open set of *A* and *U* be a nonempty open set of *X* with $A\cap U\ne \mathrm{\varnothing}$. Since *U* is an open set of *X* and $A\cap U\ne \mathrm{\varnothing}$, it follows that $U\cap A$ is a nonempty open set of *A*. As $(A,{f}_{1,\mathrm{\infty}})$ is topologically transitive, there exists $n\in \mathbb{N}$ such that ${f}_{1}^{n}({V}^{A})\cap (U\cap A)\ne \mathrm{\varnothing}$, which implies that ${f}_{1}^{n}({V}^{A})\cap U\ne \mathrm{\varnothing}$. This shows that *A* is a transitive set of $(X,{f}_{1,\mathrm{\infty}})$. □

**Theorem 3.2** *Let* $(X,{f}_{1,\mathrm{\infty}})$ *be topologically transitive and* *A* *be a regular closed set of* *X*. *Then* *A* *is a transitive set of* $(X,{f}_{1,\mathrm{\infty}})$.

*Proof* Let ${V}^{A}$ be a nonempty set of *A* and *U* be a nonempty set of *X* with $A\cap U\ne \mathrm{\varnothing}$. Since *A* is a regular closed set and $(X,{f}_{1,\mathrm{\infty}})$ is topologically transitive, there exists $n\in \mathbb{N}$ such that ${f}_{1}^{n}(int({V}^{A}))\cap U\ne \mathrm{\varnothing}$, which implies that ${f}_{1}^{n}({V}^{A})\cap U\ne \mathrm{\varnothing}$. Therefore, *A* is a transitive set of $(X,{f}_{1,\mathrm{\infty}})$. □

**Corollary 3.2** *Let* $(X,{f}_{1,\mathrm{\infty}})$ *be a non*-*autonomous discrete system*. *Then* $(X,{f}_{1,\mathrm{\infty}})$ *is topologically transitive if and only if every nonempty regular closed set of* *X* *is a transitive set of* $(X,{f}_{1,\mathrm{\infty}})$.

**Definition 3.1**Let $(X,{f}_{1,\mathrm{\infty}})$ and $(Y,{g}_{1,\mathrm{\infty}})$ be two non-autonomous discrete systems, and let $h:X\to Y$ be a continuous map and

- (1)
If $h:X\to Y$ is a surjective map, then ${f}_{1,\mathrm{\infty}}$ and ${g}_{1,\mathrm{\infty}}$ are said to be topologically semi-conjugate.

- (2)
If $h:X\to Y$ is a homeomorphism, then ${f}_{1,\mathrm{\infty}}$ and ${g}_{1,\mathrm{\infty}}$ are said to be topologically conjugate.

**Example 3.1** Let ${f}_{n}:R\to R$ with ${f}_{n}(x)=nx$ for all $n\in \mathbb{N}$ and $x\in R$, where *R* is a real line, and ${g}_{n}:{S}^{1}\to {S}^{1}$ with ${g}_{n}({e}^{i\theta})={e}^{in\theta}$ for all $n\in \mathbb{N}$, where ${S}^{1}$ is the unite circle. Define $h:R\to {S}^{1}$ by $h(x)={e}^{2\pi ix}$. It can be easily verified that *h* is a continuous surjective map and $h\circ {f}_{n}={g}_{n}\circ h$. Therefore, $(R,{f}_{1,\mathrm{\infty}})$ and $({S}^{1},{g}_{1,\mathrm{\infty}})$ are topologically semi-conjugate.

**Theorem 3.3**

*Let*$(X,{f}_{1,\mathrm{\infty}})$

*and*$(Y,{g}_{1,\mathrm{\infty}})$

*be two non*-

*autonomous discrete systems*,

*and let*$h:X\to Y$

*be a semi*-

*conjugate map*.

*A*

*is a nonempty closed subset of*

*X*

*and*$h(A)$

*is a closed subset of*

*Y*.

*Then*:

- (1)
*If**A**is a transitive set of*$(X,{f}_{1,\mathrm{\infty}})$,*then*$h(A)$*is a transitive set of*$(Y,{g}_{1,\mathrm{\infty}})$. - (2)
*If**A**is a weakly mixing set of*$(X,{f}_{1,\mathrm{\infty}})$*and*$h(A)$*is not a singleton*,*then*$h(A)$*is a weakly mixing set of*$(Y,{g}_{1,\mathrm{\infty}})$.

*Proof*(1) Let ${V}^{h(A)}$ be a nonempty open set of $h(A)$ and

*U*be a nonempty open set of

*Y*with $h(A)\cap U\ne \mathrm{\varnothing}$. Since $h(A)$ is a subspace of

*Y*, there exists an open set

*V*of

*Y*such that ${V}^{h(A)}=V\cap h(A)$. Furthermore,

*A*. Since

*A*is a transitive set of $(X,{f}_{1,\mathrm{\infty}})$, there exists $n\in \mathbb{N}$ such that ${h}^{-1}({V}^{h(A)})\cap A\cap {({f}_{1}^{n})}^{-1}({h}^{-1}(U))\ne \mathrm{\varnothing}$. As

*h*is a semi-conjugate map,

*i.e.*, ${g}_{k}(h(x))=h({f}_{k}(x))$ for every $k\in \mathbb{N}$, $x\in X$, we have ${h}^{-1}{({g}_{k})}^{-1}(x)={({f}_{k})}^{-1}{h}^{-1}(x)$ for every $k\in \mathbb{N}$, $x\in X$. Therefore, ${h}^{-1}({V}^{h(A)}\cap {({g}_{1}^{n})}^{-1}(U))\ne \mathrm{\varnothing}$, which implies that ${V}^{h(A)}\cap {({g}_{1}^{n})}^{-1}U\ne \mathrm{\varnothing}$. This shows that $h(A)$ is a transitive set of $(Y,{g}_{1,\mathrm{\infty}})$.

- (2)Suppose that
*A*is a weakly mixing set of $(X,{f}_{1,\mathrm{\infty}})$ and $h(A)$ is a closed subset of*Y*but not a singleton. Fix $k\in \mathbb{N}$. If ${V}_{1}^{h(A)},{V}_{2}^{h(A)},\dots ,{V}_{k}^{h(A)}$ are nonempty open subsets of $h(A)$ and ${U}_{1},{U}_{2},\dots ,{U}_{k}$ are nonempty open subsets of*Y*with $h(A)\cap {U}_{i}\ne \mathrm{\varnothing}$, $i=1,2,\dots ,k$. Since $h(A)$ is a subspace of*Y*, there exists an open subset ${V}_{i}$ of*Y*such that ${V}_{i}^{h(A)}={V}_{i}\cap h(A)$ for each $i=1,2,\dots ,k$. But$A\cap {h}^{-1}\left({V}_{i}^{h(A)}\right)=A\cap {h}^{-1}({V}_{i}\cap h(A))=A\cap {h}^{-1}({V}_{i}),$

*A*. Since

it follows that $A\cap {h}^{-1}({V}_{i}^{h(A)})\ne \mathrm{\varnothing}$ for each $i=1,2,\dots ,k$. Furthermore, ${h}^{-1}({U}_{i})$ is a nonempty open subset of *X* with ${h}^{-1}({U}_{i})\cap A\ne \mathrm{\varnothing}$ for each $i=1,2,\dots ,k$. Since *A* is a weakly mixing set of $(X,{f}_{1,\mathrm{\infty}})$, there exists $n\in \mathbb{N}$ such that $({h}^{-1}({V}_{i}^{h(A)})\cap A)\cap {({f}_{1}^{n})}^{-1}({h}^{-1}({U}_{i}))\ne \mathrm{\varnothing}$. As *h* is a semi-conjugate map, *i.e.*, ${g}_{m}(h(x))=h({f}_{m}(x))$ for every $m\in \mathbb{N}$, $x\in X$, we have ${h}^{-1}{({g}_{m})}^{-1}(x)={({f}_{m})}^{-1}{h}^{-1}(x)$ for every $m\in \mathbb{N}$, $x\in X$. Furthermore, ${h}^{-1}({V}_{i}^{h(A)}\cap {({g}_{1}^{n})}^{-1}{U}_{i})\ne \mathrm{\varnothing}$ for each $i=1,2,\dots ,k$, which implies that ${V}_{i}^{h(A)}\cap {({g}_{1}^{n})}^{-1}{U}_{i}\ne \mathrm{\varnothing}$ for each $i=1,2,\dots ,k$. This shows that $h(A)$ is a weakly mixing set of $(Y,{g}_{1,\mathrm{\infty}})$. □

**Corollary 3.3**

*Let*$(X,{f}_{1,\mathrm{\infty}})$

*and*$(Y,{g}_{1,\mathrm{\infty}})$

*be two non*-

*autonomous discrete systems*,

*and let*$h:X\to Y$

*be a conjugate map*.

*Then*:

- (1)
$(X,{f}_{1,\mathrm{\infty}})$

*has a transitive set if and only if so is*$(Y,{g}_{1,\mathrm{\infty}})$. - (2)
$(X,{f}_{1,\mathrm{\infty}})$

*has a weakly mixing set if and only if so is*$(Y,{g}_{1,\mathrm{\infty}})$.

**Definition 3.2** Let $(X,{f}_{1,\mathrm{\infty}})$ be a non-autonomous discrete system. ${f}_{1,\mathrm{\infty}}$ is a *k*-periodic discrete system if there exists $k\in {\mathbb{Z}}^{+}$ such that ${f}_{n+k}(x)={f}_{n}(x)$ for any $x\in X$ and $n\in {\mathbb{Z}}^{+}$.

Let $(X,{f}_{1,\mathrm{\infty}})$ be a *k*-periodic discrete system for any $k\in {\mathbb{Z}}^{+}$. Define $g=:{f}_{k}\circ {f}_{k-1}\circ \cdots \circ {f}_{1}$, we say that $(X,g)$ is an induced autonomous discrete system by a *k*-periodic discrete system $(X,{f}_{1,\mathrm{\infty}})$.

From Definition 3.2, we easily obtain the following proposition.

**Proposition 3.4**

*Let*$(X,{f}_{1,\mathrm{\infty}})$

*be a*

*k*-

*periodic non*-

*autonomous discrete system where*$(X,d)$

*is a metric space*, $g={f}_{k}\circ {f}_{k-1}\circ \cdots \circ {f}_{1}$, $(X,g)$

*is its induced autonomous discrete system*.

*Then*:

- (1)
*If*$(X,g)$*has a transitive set*,*then so is*$(X,{f}_{1,\mathrm{\infty}})$. - (2)
*If*$(X,g)$*has a weakly mixing set*,*then so is*$(X,{f}_{1,\mathrm{\infty}})$.

## Declarations

### Acknowledgements

The authors would like to thank the referees for many valuable and constructive comments and suggestions for improving this paper. This work was supported by the Education Department Foundation of Henan Province (13A110832, 14B110006), P.R. China.

## Authors’ Affiliations

## References

- Kolyada S, Snoha L: Topological entropy of nonautonomous dynamical systems.
*Random Comput. Dyn.*1996, 4: 205-233.MathSciNetMATHGoogle Scholar - Block LS, Coppel WA Lecture Notes in Mathematics 1513. In
*Dynamics in One Dimension*. Springer, Berlin; 1992.Google Scholar - Robinson C:
*Dynamical Systems: Stability, Symbolic Dynamics, and Chaos*. 2nd edition. CRC Press, Boca Raton; 1999.MATHGoogle Scholar - Walters P Texts in Math. 79. In
*An Introduction to Ergodic Theory*. Springer, New York; 1982.View ArticleGoogle Scholar - Blanchard F: Topological chaos: what may this mean?
*J. Differ. Equ. Appl.*2009, 15: 23-46. 10.1080/10236190802385355MathSciNetView ArticleMATHGoogle Scholar - Devaney RL:
*An Introduction to Chaotic Dynamical Systems*. Addison-Wesley, Redwood City; 1989.MATHGoogle Scholar - Li TY, Yorke J: Period three implies chaos.
*Am. Math. Mon.*1975, 82: 985-992. 10.2307/2318254MathSciNetView ArticleMATHGoogle Scholar - Ruette, S: Chaos for continuous interval maps: a survey of relationship between the various sorts of chaos. http://www.math.u-psud.fr/~ruette/Google Scholar
- Iwanik A: Independence and scrambled sets for chaotic mapping. In
*The Mathematical Heritage of C.F. Gauss*. World Scientific, River Edge; 1991:372-378.View ArticleGoogle Scholar - Blanchard F, Huang W: Entropy sets, weakly mixing sets and entropy capacity.
*Discrete Contin. Dyn. Syst.*2008, 20: 275-311.MathSciNetMATHGoogle Scholar - Xiong J, Yang Z: Chaos caused by a topologically mixing map. Advanced Series in Dynamical Systems 9. In
*Dynamical Systems and Related Topics*. World Scientific, Singapore; 1991:550-572.Google Scholar - Oprocha P, Zhang G: On local aspects of topological weak mixing in dimension one and beyond.
*Stud. Math.*2011, 202: 261-288. 10.4064/sm202-3-4MathSciNetView ArticleMATHGoogle Scholar - Kolyada S, Snoha L, Trofimchuk S: On minimality of nonautonomous dynamical systems.
*Nonlinear Oscil.*2004, 7: 83-89.MathSciNetView ArticleGoogle Scholar - Kempf R: On Ω-limit sets of discrete-time dynamical systems.
*J. Differ. Equ. Appl.*2002, 8: 1121-1131. 10.1080/10236190290029024MathSciNetView ArticleMATHGoogle Scholar - Canovas JS: On
*ω*-limit sets of non-autonomous discrete systems.*J. Differ. Equ. Appl.*2006, 12: 95-100. 10.1080/10236190500424274MathSciNetView ArticleMATHGoogle Scholar - Krabs W: Stability and controllability in non-autonomous time-discrete dynamical systems.
*J. Differ. Equ. Appl.*2002, 8: 1107-1118. 10.1080/1023619021000053971MathSciNetView ArticleMATHGoogle Scholar - Huang X, Wen X, Zeng F: Topological pressure of nonautonomous dynamical systems.
*Nonlinear Dyn. Syst. Theory*2008, 8: 43-48.MathSciNetMATHGoogle Scholar - Huang X, Wen X, Zeng F: Pre-image entropy of nonautonomous dynamical systems.
*J. Syst. Sci. Complex.*2008, 21: 441-445. 10.1007/s11424-008-9125-7MathSciNetView ArticleMATHGoogle Scholar - Shi Y, Chen G: Chaos of time-varying discrete dynamical systems.
*J. Differ. Equ. Appl.*2009, 15: 429-449. 10.1080/10236190802020879View ArticleMathSciNetMATHGoogle Scholar - Oprocha P, Wilczynski P: Chaos in nonautonomous dynamical systems.
*An. Univ. ‘Ovidius’ Constanţa, Ser. Mat.*2009, 17: 209-221.MathSciNetMATHGoogle Scholar - Canovas JS: Li-Yorke chaos in a class of non-autonomous discrete systems.
*J. Differ. Equ. Appl.*2011, 17: 479-486. 10.1080/10236190903049025MathSciNetView ArticleMATHGoogle Scholar - Kuang R, Cheng W: Li-Yorke chaos in a class of non-autonomous discrete systems.
*J. Differ. Equ. Appl.*2011, 17: 479-486. 10.1080/10236190903049025View ArticleGoogle Scholar - Dickman RF Jr.: Regular closed maps.
*Proc. Am. Math. Soc.*1973, 39: 414-416. 10.1090/S0002-9939-1973-0315654-1MathSciNetView ArticleMATHGoogle Scholar - Engelking R:
*General Topology*. Panstwowe Wydawnictwo Naukove, Warsaw; 1977.MATHGoogle Scholar

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