Localization of Δ-Mixing Property via Furstenberg Families

Abstract

This paper is devoted to studying the localization of -mixing property via Furstenberg families. The notion of -weakly mixing sets is extended to --weakly mixing sets with respect to a sequence, and the characterization of -weakly mixing sets is also generalized.

1. Introduction

Throughout this paper, a topological dynamical system (or dynamical system, system for short) is a pair , where is a compact metric space with metric and is a continuous map of to itself.

In some literature on nonautonomous discrete systems, the topological dynamical system is usually called the autonomous discrete system; for example, see [1].

Let be a dynamical system. For two subsets , of , we define the hitting time set of and by We say that is transitive if, for every two nonempty open subsets and of , the hitting time set is nonempty, weakly mixing if the product system is transitive, and strongly mixing if, for every two nonempty open subsets and of , there exists such that . In his seminal paper [2], Furstenberg showed that weak mixing implies -fold transitivity for every positive integer as follows. (i)Let be a topological dynamical system. Then is transitive if and only if, for every , the product system (-times) is also transitive.

For convenience, we denote (-times) by .

A closed subset of is called -transitive if, for every , there exists a residual subset of such that, for every , the orbit closure of the diagonal -tuple , that is, , under the action contains . A closed subset of with at least two points is -weakly mixing if, for every , is -transitive in the -th product system . In [3], Huang et al. show that -weakly mixing sets exhibit a nice characterization as follows.

Theorem 1 (see [3]). Let be a dynamical system and a closed subset of but not a singleton. Then is -weakly mixing if and only if there exists a strictly increasing sequence of Cantor sets of such that is dense in and (i)for any , any subset of , and any continuous functions for , there exists a strictly increasing sequence of positive integers such that  for every and ;(ii)for any , , any closed subset of , and continuous functions for , there exists a strictly increasing sequence of positive integers such that  uniformly on and .

In this paper, we will extend the notion of -weakly mixing sets via Furstenberg family and show that --weakly mixing sets with respect to a sequence also share the same characterization under some considerable conditions.

2. Preliminary

In this section, we provide some definitions which will be used later.

Let denote the collection of all subsets of . A subset of is called a Furstenberg family (or family for short ), if it admits the property of hereditary upward; that is to say, A family is called proper if it is neither empty nor all of .

For a family , the dual family of is defined by Let be the family of all infinite subsets of . It is easy to see that its dual family , denoted by , is the family of all cofinite subsets.

Any nonempty collection of subsets of naturally generates a family We say that is countable generated if for a collection consisting of countable subsets of .

The idea of using families to describe dynamical properties goes back at least to Gottschalk and Hedlund [4] and was developed further by Furstenberg [5]. For recent results, see [6–9].

Let be a dynamical system, a family, and a strictly increasing sequence of nonnegative integers with .

For every and subsets of , we define the hitting time set of by

We say that is -weakly mixing with respect to if, for every , and nonempty open subsets of intersecting for and , we have

Let be a compact metric space. Denote by the collection of all nonempty closed subsets of and endow with the Hausdorff metric for any Then the metric space is compact whenever is compact. For any nonempty subsets , denote then the following family forms a basis for a topology of , which is called the Vietoris topology. It is well known that the Hausdorff topology induced by the Hausdorff metric coincides with the Vietoris topology for .

3. Key Lemmas

In this section, we provide some lemmas which will be used later.

Let be a dynamical system, a family, a strictly increasing sequence of nonnegative integers with , and a closed subset of . For , , and any , we say that a subset of is -spread with respect to , if there exist , , and distinct points such that and for any maps where , there exists such that and for .

For any , denote by the collection of all closed sets that are -spread in with respect to .

Remark 2. It is not hard to check that the is hereditary; that is, if is -spread in with respect to and is a nonempty closed subset of , then is also -spread in with respect to .

Let

Lemma 3. If is a -weakly mixing subset of but not a singleton, then is a dense open subset of for every .

Proof. Fix , , and . We will divide our discussion into three claims to show that is a dense open subset of .
Claim 4. is open in .
Proof of Claim 4. Let . For any , let and as in the definition of set -spread in with respect to . For , put . Then, by Remark 2, every closed subset is -spread in with respect to . It follows that Hence is open in .□
Claim 5. is a perfect set.
Proof of Claim 5. If is not perfect, then there exists an isolated point of . Note that the set is open in . It follows that there are two nonempty open subsets such that and since the set is not a singleton. So, for any , such that , one has . This is a contradiction, and thus is perfect.□
Claim 6. is dense in .
Proof of Claim 6. Fix nonempty open subsets of intersecting . We want to show that Since is compact, there exists a finite subset of such that . For convenience, denote for . Since is --weakly mixing with respect to , the set is perfect, so we may assume that . The collection of -tuples on the set can be arranged as the following finite sequence: For , as is --weakly mixing with respect to , there exists with , such that for . By continuity of , we can choose a nonempty subset of intersecting such that for . Similarly, for , since is a strictly increasing sequence of positive integers, there exist with and nonempty subsets of intersecting such that for and . After repeating this process times, we obtain distinct positive integers with , and nonempty open subsets intersecting such that , andfor , and .
For each , pick . Since is perfect, it is reasonable to assume that those ’s are distinct. It is clear that Choose such that for . For any map for , there exists an -tuple such that for , and . Thus there is () such that for , and . This implies is -spread in with respect to and hence is dense in .□

Lemma 7. Let be a dynamical system and a Furstenberg family with being countable generated. If is a -weakly mixing subset with respect to but not a singleton, then is a residual subset of .

Proof. Since is countable generated, there exists a sequence of such that By Lemma 3, it follows that And this completes the proof of Lemma 7.

Lemma 8. If , then, for any closed subset of , , and continuous functions for , there exists a strictly increasing sequence such that uniformly on and .

Proof. Let be a closed subset of , . Let , , be continuous functions. Then is uniform continuous and this follows that, for any , there exists such that Choose with . Let and be as in the definition of which is -spread in . By the definition of -spread subset, there exists with such that for and . Without lose of generality, it can be assumed that is increasing.
We are going to show that the sequence is as required.
For any , there exists such that . Then for . Thus for . This completes the proof of Lemma 8.

Lemma 9. If is a strictly increasing sequence of sets in , then, for any subset A of , and continuous functions for , there exists a strictly increasing sequence such that for every and .

Proof. Let , , and , , be continuous functions. For , take . Since is hereditary, the closure of is also in for all .
For any , the set is -spread in . Let and be as in the definition of which is -spread in . Then there exists with such that for . Without lose of generality, it can be assumed that is increasing.
We are going to show that the sequence is as required.
Fix any . There exists such that for all . By the continuity of , for any , there exists with such that For every , there exists such that . Then for . Thus for . This completes the proof of Lemma 9.

4. The Characterization of --Mixing Sets

In this section, we will show the following theorem which generalizes the result in [3].

Theorem 10. Let be a dynamical system, a Furstenberg family with being countable generated, a closed subset of but not a singleton, and a strictly increasing sequence of nonnegative integers with . Then is -weakly mixing with respect to if and only if, for any , there exists an increasing sequence of Cantor sets of such that is dense in and (i)for any , any subset of , and any continuous functions for , there exists a strictly increasing sequence of such that  for every and ;(ii)for any , , any closed subset of , and continuous functions for , there exists a strictly increasing sequence of such that  uniformly on and .

A subset of is called hereditary if for every set . The following lemma, which is a consequence of the Kuratowski-Mycielski Theorem, is cited from [3].

Lemma 11. Let be a perfect compact space. If a hereditary subset of is residual then there exists an increasing sequence of Cantor sets of such that (i) for every ;(ii) is dense in .

Proof of Theorem 10.  
Necessity. Since is --weakly mixing with respect to , by Lemma 7, is a residual subset of . Now, by Lemma 11, there exists a strictly increasing sequence of Cantor sets of such that for every and is dense in . The conclusion follows from Lemmas 8 and 9.
Sufficiency. Fix any . Let be the set satisfying the requirement. Let and be nonempty open subsets of intersecting for , and . It is not hard to see that is perfect. It follows that there exist pairwise distinct points for , and .
For , define by for . Choose such that for all . It is clear that are continuous; thus we can find such that for all . Then for all , which implies Therefore, the set is --weakly mixing with respect to .

It is not hard to see that a subset of is -weakly mixing if and only if --weakly mixing with repect to , and since , where for each , it follows that is coutable generated. Thus our result extended [3, Theorem A].

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors thank the referee and the editor for their valuable suggestions, and the authors sincerely thank Doctor Zhi-Jing Chen for his kind help and concern all the time. The second and third authors are supported by NSF of Anhui Province (no. 1608085QA12), NSF of Education Committee of Anhui Province (nos. KJ2016A506 and KJ2017A454), Excellent Young Talents Foundation of Anhui Province (no. GXYQ2017070), Doctoral Scientific Research Foundation of Chaohu University (no. KYQD-201605), and Scientific Research Project of Chaohu University (no. XLY-201501).