 Research
 Open Access
 Published:
Weakly mixing sets and transitive sets for nonautonomous discrete systems
Advances in Difference Equations volume 2014, Article number: 217 (2014)
Abstract
In this paper we mainly study the weakly mixing sets and transitive sets of nonautonomous discrete systems. Some basic concepts are introduced for nonautonomous discrete systems, including a weakly mixing set and a transitive set. We discuss the basic properties of weakly mixing sets and transitive sets of nonautonomous discrete systems. Also, we investigate the relationship between two conjugated nonautonomous 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 nonautonomous discrete system [1]. Define
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
Its longterm 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; LiYorke chaos (see [7]) and positive entropy (see [2, 8]) belong to partial properties. Weak mixing is an overall property, it is stable under semiconjugate maps and implies LiYorke 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 LiYorke chaos’ and ‘LiYorke 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 nonautonomous discrete systems. Kolyada and Snoha [1] gave the definition of topological entropy in nonautonomous discrete systems; Kolyada et al. [13] discussed minimality of nonautonomous discrete systems; Kempf [14] and Canovas [15] studied ωlimit sets in nonautonomous discrete systems. Krabs [16] discussed stability in nonautonomous discrete systems; Huang et al. [17, 18] studied topological pressure and preimage entropy of nonautonomous discrete systems. Shi and Chen [19] and Oprocha and Wilczynski [20] and Canovas [21] discussed chaos in nonautonomous discrete systems, respectively. Kuang and Cheng [22] studied fractal entropy of nonautonomous 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 nonautonomous discrete system. We discuss the basic properties of weakly mixing sets and transitive sets for nonautonomous discrete systems. Moreover, we investigate the weakly mixing sets and transitive sets for the conjugated nonautonomous 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 ${f}_{n}\circ {f}_{n1}\circ \cdots \circ {f}_{2}\circ {f}_{1}$, i.e., ${f}_{1}^{n}={f}_{n}\circ {f}_{n1}\circ \cdots \circ {f}_{2}\circ {f}_{1}$ for any $n\in \mathbb{N}$. We define
for any $k,n\in \mathbb{N}$.
A nonautonomous 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 nonautonomous 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 nonautonomous 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 nonautonomous 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 nonautonomous 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 $[0,1]$.
Observe that the given sequence converges uniformly to the tent map
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.
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 nonautonomous discrete systems.
Proposition 3.1 Let $(X,{f}_{1,\mathrm{\infty}})$ be a nonautonomous 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 nonautonomous 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 nonautonomous 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,
$${\left({f}_{1}^{n}\right)}^{1}(U)={\left({f}_{1}^{n}\right)}^{1}(X\setminus E)={\left({f}_{1}^{n}\right)}^{1}(X)\setminus {\left({f}_{1}^{n}\right)}^{1}(E)\subseteq X\setminus E=U\phantom{\rule{1em}{0ex}}\text{for every}n\in \mathbb{N}.$$
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 nonautonomous 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 nonautonomous 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 nonautonomous 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 semiconjugate.

(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 semiconjugate.
Theorem 3.3 Let $(X,{f}_{1,\mathrm{\infty}})$ and $(Y,{g}_{1,\mathrm{\infty}})$ be two nonautonomous discrete systems, and let $h:X\to Y$ be a semiconjugate 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,
Hence, $A\cap {h}^{1}({V}^{h(A)})$ is an open subset of A. Since
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 semiconjugate 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
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 semiconjugate 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 nonautonomous 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 nonautonomous discrete system. ${f}_{1,\mathrm{\infty}}$ is a kperiodic 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 kperiodic discrete system for any $k\in {\mathbb{Z}}^{+}$. Define $g=:{f}_{k}\circ {f}_{k1}\circ \cdots \circ {f}_{1}$, we say that $(X,g)$ is an induced autonomous discrete system by a kperiodic 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 kperiodic nonautonomous discrete system where $(X,d)$ is a metric space, $g={f}_{k}\circ {f}_{k1}\circ \cdots \circ {f}_{1}$, $(X,g)$ 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 $(X,{f}_{1,\mathrm{\infty}})$.
References
 1.
Kolyada S, Snoha L: Topological entropy of nonautonomous dynamical systems. Random Comput. Dyn. 1996, 4: 205233.
 2.
Block LS, Coppel WA Lecture Notes in Mathematics 1513. In Dynamics in One Dimension. Springer, Berlin; 1992.
 3.
Robinson C: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. 2nd edition. CRC Press, Boca Raton; 1999.
 4.
Walters P Texts in Math. 79. In An Introduction to Ergodic Theory. Springer, New York; 1982.
 5.
Blanchard F: Topological chaos: what may this mean? J. Differ. Equ. Appl. 2009, 15: 2346. 10.1080/10236190802385355
 6.
Devaney RL: An Introduction to Chaotic Dynamical Systems. AddisonWesley, Redwood City; 1989.
 7.
Li TY, Yorke J: Period three implies chaos. Am. Math. Mon. 1975, 82: 985992. 10.2307/2318254
 8.
Ruette, S: Chaos for continuous interval maps: a survey of relationship between the various sorts of chaos. http://www.math.upsud.fr/~ruette/
 9.
Iwanik A: Independence and scrambled sets for chaotic mapping. In The Mathematical Heritage of C.F. Gauss. World Scientific, River Edge; 1991:372378.
 10.
Blanchard F, Huang W: Entropy sets, weakly mixing sets and entropy capacity. Discrete Contin. Dyn. Syst. 2008, 20: 275311.
 11.
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; 1991:550572.
 12.
Oprocha P, Zhang G: On local aspects of topological weak mixing in dimension one and beyond. Stud. Math. 2011, 202: 261288. 10.4064/sm20234
 13.
Kolyada S, Snoha L, Trofimchuk S: On minimality of nonautonomous dynamical systems. Nonlinear Oscil. 2004, 7: 8389.
 14.
Kempf R: On Ωlimit sets of discretetime dynamical systems. J. Differ. Equ. Appl. 2002, 8: 11211131. 10.1080/10236190290029024
 15.
Canovas JS: On ω limit sets of nonautonomous discrete systems. J. Differ. Equ. Appl. 2006, 12: 95100. 10.1080/10236190500424274
 16.
Krabs W: Stability and controllability in nonautonomous timediscrete dynamical systems. J. Differ. Equ. Appl. 2002, 8: 11071118. 10.1080/1023619021000053971
 17.
Huang X, Wen X, Zeng F: Topological pressure of nonautonomous dynamical systems. Nonlinear Dyn. Syst. Theory 2008, 8: 4348.
 18.
Huang X, Wen X, Zeng F: Preimage entropy of nonautonomous dynamical systems. J. Syst. Sci. Complex. 2008, 21: 441445. 10.1007/s1142400891257
 19.
Shi Y, Chen G: Chaos of timevarying discrete dynamical systems. J. Differ. Equ. Appl. 2009, 15: 429449. 10.1080/10236190802020879
 20.
Oprocha P, Wilczynski P: Chaos in nonautonomous dynamical systems. An. Univ. ‘Ovidius’ Constanţa, Ser. Mat. 2009, 17: 209221.
 21.
Canovas JS: LiYorke chaos in a class of nonautonomous discrete systems. J. Differ. Equ. Appl. 2011, 17: 479486. 10.1080/10236190903049025
 22.
Kuang R, Cheng W: LiYorke chaos in a class of nonautonomous discrete systems. J. Differ. Equ. Appl. 2011, 17: 479486. 10.1080/10236190903049025
 23.
Dickman RF Jr.: Regular closed maps. Proc. Am. Math. Soc. 1973, 39: 414416. 10.1090/S00029939197303156541
 24.
Engelking R: General Topology. Panstwowe Wydawnictwo Naukove, Warsaw; 1977.
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.
Author information
Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
LL (the first author) carried out the study of weakly mixing sets and transitive sets for nonautonomous discrete systems and drafted the manuscript. YS (the second author) helped to draft the manuscript. All authors read and approved the final manuscript.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Liu, L., Sun, Y. Weakly mixing sets and transitive sets for nonautonomous discrete systems. Adv Differ Equ 2014, 217 (2014). https://doi.org/10.1186/168718472014217
Received:
Accepted:
Published:
Keywords
 nonautonomous discrete system
 weakly mixing set
 transitive set