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

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.

1 Introduction

Throughout this paper ℕ denotes the set of all positive integers, and let ${\mathbb{Z}}_{+}=\mathbb{N}\cup \{0\}$. Let 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

$$f_1^n(x):=f_n\circ f_{n-1}\circ \cdots \circ f_2\circ f_1,n\in \mathbb{N},$$

and $f_1^0:={\mathrm{id}}_X$, the identity on 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

$$\gamma (x,f_{1,\mathrm{\infty }})=\{x,f_1(x),f_1^2(x),\dots ,f_1^n(x),\dots \}.$$

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

In the present paper, $\overline{A}$ and $int(A)$ denote the closure and interior of the set 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

$${(f_k)}^n=\underset{n}{\underbrace{f_k\circ f_k\circ \cdots \circ f_k}}$$

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

$$f_n(x)=\{\begin{array}{ll}\frac{2n}{n-2}x& \text{if }0\le x\le \frac{n-2}{2n},\\ 1& \text{if }\frac{n-2}{2n}\le x\le \frac{n+2}{2n},\\ \frac{2n}{n-2}(1-x)& \text{if }\frac{n+2}{2n}\le x\le 1,\end{array}n=3,4,\dots$$

and $f_1=f_2=\mathrm{id}$, the identity on $[0,1]$.

Observe that the given sequence converges uniformly to the tent map

$$f(x)=\{\begin{array}{ll}2x& \text{if }0\le x\le \frac{1}{2},\\ 2(1-x)& \text{if }\frac{1}{2}\le x\le 1,\end{array}$$

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 $(X,f_{1,\mathrm{\infty }})$.

Figure 1 and Figure 2 denote the tent map f and the 2nd iterate $f^2$ of the tent map f, respectively.

Figure 1

The tent map f .

Figure 2

The 2nd iterate ${\mathit{f}}^{\mathbf{2}}$ of the tent map f .

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

  $$V^A\cap \bigcup _{n\in \mathbb{N}}{\left(f_1^n\right)}^{-1}U=\bigcup _{n\in \mathbb{N}}(V^A\cap {\left(f_1^n\right)}^{-1}(U))\ne \mathrm{\varnothing }.$$

Therefore, ${\bigcup }_{n\in \mathbb{N}}{(f_1^n)}^{-1}(U)$ is dense in 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,

  $${\left(f_1^n\right)}^{-1}(U)={\left(f_1^n\right)}^{-1}(X\setminus E)={\left(f_1^n\right)}^{-1}(X)\setminus {\left(f_1^n\right)}^{-1}(E)\subseteq X\setminus E=U\text{for every }n\in \mathbb{N}.$$

By the result of (1), 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

$$g_n(h(x))=h(f_n(x))\text{for any }n\in \mathbb{N},x\in X.$$

• (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,

$$A\cap h^{-1}\left(V^{h(A)}\right)=A\cap h^{-1}(V\cap h(A))=A\cap h^{-1}(V).$$

Hence, $A\cap h^{-1}(V^{h(A)})$ is an open subset of A. Since

$$h(A\cap h^{-1}\left(V^{h(A)}\right))=h(A)\cap V^{h(A)}=V^{h(A)}\ne \mathrm{\varnothing },$$

then we have $A\cap h^{-1}(V^{h(A)})\ne \mathrm{\varnothing }$. Moreover, $U\cap h(A)\ne \mathrm{\varnothing }$, which implies that $h^{-1}(U)\cap A\ne \mathrm{\varnothing }$. 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

  $$A\cap h^{-1}\left(V_i^{h(A)}\right)=A\cap h^{-1}(V_i\cap h(A))=A\cap h^{-1}(V_i),$$

then $A\cap h^{-1}(V_i^{h(A)})$ ($i=1,2,\dots ,k$) are open subsets of A. Since

$$h(A\cap h^{-1}\left(V_i^{h(A)}\right))=h(A)\cap V_i^{h(A)}=V_i^{h(A)}\ne \mathrm{\varnothing },$$

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 $(X,f_{1,\mathrm{\infty }})$.