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Weakly mixing, topologically weakly mixing, and sensitivity for non-autonomous discrete systems

This paper is concerned with relationships of weakly mixing, topologically weakly mixing, and sensitivity for non-autonomous discrete systems. It is shown that weakly mixing implies topologically weakly mixing and sensitivity for measurable systems with a fully supported measure; and topological weakly mixing implies sensitivity for general dynamical systems. However, the inverse conclusions are not true and some counterexamples are given. The related existing results for autonomous discrete systems are generalized to non-autonomous discrete systems and their conditions are weaken.


Introduction
Chaos is a universal dynamical behavior of nonlinear dynamical systems and one of the central topics of research on nonlinear science. It is well known that sensitivity characterizes the unpredictability of chaotic phenomena, and is the essential condition of various definitions of chaos. Therefore, the study on sensitivity has attracted a lot of attention from many scholars [1,2,5,7,8,10,12].
In 2002, Abraham et al. proved that if a measure-preserving map f on a metric probability space X with a fully supported measure is either topologically mixing or weakly mixing, and satisfies that for any nonempty open set U ⊂ X, there is a sequence {n k } ∞ k=0 with positive upper density such that U ∩ (∩ k≥0 f −n k U) = ∅, then f is sensitive [1]. In 2004, He et al. relaxed the conditions of the above result and showed that if a measure-preserving map (resp. a measure-preserving semi-flow) on a metric probability space with a fully supported measure is weakly mixing, then it is sensitive [8]. In 2010, Li and the second author of the present paper gave several sufficient conditions of sensitivity for maps and semi-flows, and showed that if a measure-preserving map (resp. a measurepreserving semi-flow) on a metric probability space with a fully supported measure is topologically strongly ergodic, then it is sensitive [12].
Since many complex systems occurring in the real-world problems such as physical, biological, and economical problems are necessarily described by non-autonomous discrete systems, which are generated by iteration of a sequence of maps in a certain order, many scientists and mathematicians focused on complexity of non-autonomous discrete systems recently [3,6,10,11,[14][15][16][17][18]20]. In 1996, Kolyada and Snoha introduced the concept of topological entropy for a non-autonomous discrete system and studied its properties [11]. In 2006, Tian and Chen extended the concept of chaos in the sense of Devaney to non-autonomous discrete systems [18]. In 2009, related concepts of chaos, such as topological transitivity, sensitivity, chaos in the sense of Li-Yorke, Wiggins, and Devaney were extended to general non-autonomous discrete systems [15]. Recently, some sufficient conditions of sensitivity for general non-autonomous discrete systems are presented in [10]. Motivated by the above results, we shall try to investigate the relationships of weakly mixing, topologically weakly mixing, and sensitivity for non-autonomous discrete systems in the present paper.
The rest of the paper is organized as follows. In Section 2, some basic concepts and useful lemmas are presented. In Section 3, relationships of weakly mixing, topologically weakly mixing, and sensitivity for non-autonomous discrete systems are discussed.

Preliminaries
In this section, some basic concepts for non-autonomous discrete systems are introduced, including topological transitivity, topologically weakly mixing, topologically mixing, weakly mixing, and sensitivity. In addition, some useful lemmas are also presented. We shall consider the following non-autonomous discrete system in the present paper: where f n : X → X is a map for each n ≥ 0 and (X, d) is a metric space.
For any fixed x 0 ∈ X, {x n } ∞ n=0 is called the (positive) orbit of system (2.1) starting from x 0 and  Note that topologically mixing implies topologically weakly mixing, and topologically weakly mixing implies topological transitivity for system (2.1) in A.
Let B(X) be the σ-algebra of Borel subsets of X, and µ be a finite measure of measurable space (X, B(X)).   Proof. It can be directly derived by Lemma 2.1.

Relationships of weakly mixing, topologically weakly mixing, and sensitivity
In this section, we shall investigate some relationships of weakly mixing, topologically weakly mixing, and sensitivity for system (2.1).
Theorem 3.1. If system (2.1) is topologically weakly mixing, then it is sensitive in X.
Assertion (ii) holds by assertion (i) and Theorem 3.1. This completes the proof.
Remark 3.2. Assertion (ii) in Theorem 3.2 can also be proved in a direct way using Theorem 3.1 in [10]. By the results of Theorems 3.1-3.3, one gets that weakly mixing implies topologically weakly mixing for measurable systems with a fully supported measure, and topologically weakly mixing implies sensitivity for general dynamical systems. It is natural to ask whether their reverse conclusions hold. That is, does topologically weakly mixing imply weakly mixing? and does sensitivity imply topologically weakly mixing? For the first question, one counterexample has been given, see [4, p. 32] and [19]. For the second question, a counterexample is given as follows. Clearly, f maps X into itself. Since f n ((0, 1)) ∩ (1, 3/2) = ∅ for each n ≥ 1, f is not topologically transitive in X and thus it is not topologically weakly mixing in X. On the other hand, one can easily verify that f is sensitive in X since the tent map is sensitive in [0, 1], and f (1, 3/2] ⊂ [0, 1]. Therefore, f is sensitive in X.