The Topological Sensitivity with respect to Furstenberg Families

Wang, Tengfei; Jing, Kai; Yin, Jiandong

doi:https://doi.org/10.1155/2020/7684072

Discrete Dynamics in Nature and Society

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Research Article | Open Access

Volume 2020 | Article ID 7684072 | https://doi.org/10.1155/2020/7684072

The Topological Sensitivity with respect to Furstenberg Families

Tengfei Wang,¹Kai Jing,¹and Jiandong Yin¹

Academic Editor: Cengiz Çinar

Received20 Feb 2020

Accepted16 May 2020

Published04 Jun 2020

Abstract

In this work, a dynamical system means that is a topological space and is a continuous map. The aim of the article is to introduce the conceptions of topological sensitivity with respect to Furstenberg families, -topological sensitivity, and multisensitivity and present some of their basic features and sufficient conditions for a dynamical system to possess some sensitivities. Actually, it is proved that every topologically ergodic but nonminimal system is syndetically sensitive and a weakly mixing system is -thickly topologically sensitive and multisensitive under the assumption that admits some separability.

1. Introduction

For a compact system which means that is a continuous self-map on a compact metric space , sensitive dependence on initial condition (sensitive for simplicity) for was firstly introduced by Ruelle [1] as if there exists such that for each and every open neighborhood of , there is a nonnegative integer such that . One can write this in a slightly different way (see [2]) as follows. For a nonempty and , letwhere denotes the set of nonnegative integers. Then, a compact system is sensitive if and only if there exists such that for each nonempty open (opene for simplicity) subset of .

Sensitivity is a key conception used to characterize the unpredictability of a compact system and a chief component of some chaotic properties such as the chaos in the sense of Devaney [3] and Banks et al. [4]. In [5, 6], the authors introduced the linear chaos and linear topological dynamics, and one can in [7] for the concept of multivalued linear and nonlinear topological dynamics. More detailed information of the related studies of sensitivity are introduced in [2, 8–13] and [14]. For the sake of distinguishing the following topological version of sensitivity, we use the classical sensitivity to stand for the sensitivity of compact systems in this paper.

From now on, we call the pair a dynamical system if is a continuous self-map on a topological space . In [15], the author introduced the topological version of sensitivity (topological sensitivity for short) for dynamical systems as follows.

Definition 1. (see [15]). Let be a dynamical system. An open cover of is called a sensitivity cover (-cover for short) for if for every opene subset of there exist and such that .

Definition 2. (see [15]). A dynamical system (or simply the map ) is called topologically sensitive if has an -cover.

In other words, a dynamical system is topologically sensitive if it admits an open cover satisfying that, for every opene set in , there are and such that for each .

Topological sensitivity of dynamical systems generalizes the classical sensitivity of compact systems since there exist in [5] dynamical systems on metric noncompact spaces which are topologically sensitive but not classically sensitive. The author of [15] presented some sufficient conditions for a dynamical system to be topologically sensitive. For example, it was proven in [15] that each transitive map possessing a dense set of almost periodic points and admitting an eventually periodic point on an infinite Urysohn space is topologically sensitive; every weakly mixing map is topologically sensitive if there are two opene subsets in the phase space such that the intersection of their closures is empty. Moreover, the weakly positively expansive maps were also considered in [15].

In this paper, for the sake of dealing with the conception of topological sensitivity of dynamical systems in a unified way, we introduce the conceptions of topological sensitivity with respect to families of , -topological sensitivity, and multisensitivity for dynamical systems and give some sufficient conditions for a dynamical system to possess distinct sensitivities. Some of the presented results improve or generalize the main results of [15] to a great extent.

2. Preliminaries

In this section, we recall some notations, notions and basic theories of nonnegative integers and dynamical systems.

2.1. Subsets of Nonnegative Integers

Throughout this paper, denote by the set of nonnegative integers, the set of positive integers, the set of integers, and the set of real numbers, respectively.

Let be the collection of all subsets of . A subset of is called a Furstenberg family (family for short) of provided it is hereditary upward, that is, and imply . A family of is proper if it is a proper subset of , namely, and ; translation invariant if for each , for each ; and a filter if it is proper and closed under intersection, i.e., implies .

Let be a family of , write , namely, is the smallest family generated by the family .

A set is called thick if for each there exists such that and syndetic if there exists such that, for every , . By their definitions, it is obvious that every thick subset of intersects each syndetic subset of .

A set is called piecewise syndetic if it can be written as the intersection of a thick set and a syndetic set and thickly syndetic if for any , there exists a syndetic set such that

The upper density of a subset of is defined aswhere denotes the cardinality of the set . Similarly one can define the lower density of .

The upper Banach density of is defined asThe supremun is taken over all segments of . One can see [16] for more details of families.

In general, we use to denote the family consisting of all infinite subsets of and use , , , , , , and to denote, respectively, the families of syndetic subsets, thick subsets, thickly syndetic subsets, piecewise syndetic subsets, the subsets with positive upper density, the subsets with positive lower density, and the subsets with positive upper Banach density of . Then, , , , , , , and are proper and translation invariant. About the sets with positive upper Banach density, there is a result in [16] stated as follows which will be used in the proof of Lemma 2 in the paper.

Proposition 1. (see [16]). For each , is syndetic. Here, .

2.2. Basic Notions of Dynamical Systems

Let be a dynamical system. Let denote the closure of in and denotes the difference set of , i.e., .

For nonempty open (opene in brief and hereinafter) subsets of and , let denote the collection of all open neighborhoods of and write and .

A point is a transitive point of if , the orbit of under , is dense in . A dynamical system is minimal if each point in is transitive. In a dynamical system , is called a minimal point if the dynamical system is minimal. A subset of is called minimal if every point of is minimal.

A point is called an almost periodic point of if for each . Denote by the set of all almost periodic points of . In [17], the authors introduced the concept of positive upper Banach density recurrent points for compact systems. Now, we introduce this notion for dynamical systems.

Definition 3. Let be a dynamical system. A point is called a positive upper Banach density recurrent point of if for each .

Denote by the set of all positive upper Banach density recurrent points of . Obviously, .

Let be a family of . A dynamical system is -transitive if for each pair of opene subsets of , ; -central if for each opene subset of , . Especially, a dynamical system is transitive if for each pair of opene subsets of , ; topological ergodic if for each pair of opene subsets of ; weakly mixing if the product system is transitive. In Section 3, we will prove that a weakly mixing system possesses some analogous properties for the hitting time set of any two opene subsets to compact systems.

A dynamical system is called an -system if it is transitive and the set of its minimal points is dense in .

A dynamical system is semiconjugate to a dynamical system if there exists a continuous surjection such that . Meanwhile, is called a semiconjugation from to . Moreover, a semiconjugation from to is semiopen if has a nonempty interior for each opene . Especially, if is a homeomorphism from to , then is said to be conjugate to .

Now, based on Definitions 1 and 2, we introduce a more general version of topological sensitivity for a dynamical system. Actually, we introduce the notion of topological sensitivity with respect to a family of stated as below.

Definition 4. Let be a dynamical system, be a family of , and be an open cover of . is called an -sensitivity cover for if for each opene subset of ,

Now, by using the notion of -sensitivity covers, we introduce the topological version of sensitivity with respect to a Furstenberg family for dynamical systems.

Definition 5. Let be a family of . A dynamical system is called -topologically sensitive if admits an -sensitivity cover.

Obviously, a dynamical system is topologically sensitive if and only if it is -topologically sensitive with respect to the family and the notion of {family}-topological sensitivity of dynamical systems generalizes that of {family}-sensitivity of compact systems.

Example 1. Let be the self-map on defined by , and let us consider the following metric on : . Then, this metric is equivalent to the usual metric on , namely, they generate the same topology of . From [15], it is known that is topologically sensitive. In fact, it is not hard to check that is -topologically sensitive for the families .
Take an open cover of . For any opene subset of and with , let . Obviously, each is cofinite, i.e., is finite. Put . For any opene subset of and with and , we haveThat is, for each and . If not, there exist some opene subset of and with and and such that and ; then,On the contrary,which is a contradiction. Therefore, for every opene set and every pair with , and . It implies that is an -sensitivity cover of .

3. Topological Sensitivity with respect to Families of Dynamical Systems

In this section, we give some basic properties of -topological sensitivity and prove that the -topological sensitivity is invariant under semiconjugations and that a dynamical system is -topologically sensitive (resp. , , , , , -topologically sensitive) if and only if so is for each if and only if so is for some .

Theorem 1. Suppose that and are two dynamical systems and is a family of . If and are semiconjugate and the semiconjugation from to is semiopen and is -topologically sensitive, then so is .

Proof. Let be an -sensitivity cover for . Set , then is an open cover of . For each opene subset of , has a nonempty interior since is semiopen. Take an opene set . By the -topological sensitivity of ,Thus, for each , there exist such that, for every , or . Take such that and . Then, for every , or . So, . The definition of families implies for each opene subset of . Therefore, is -topologically sensitive.

In the following, we will show that the -topological sensitivity as well as , , , , , and -topological sensitivity of a dynamical system is invariant under iterations.

Theorem 2. Let be a dynamical system. Then, the following statements are equivalent:(1) is -topologically sensitive (resp. , , -topologically sensitive)(2) is -topologically sensitive (resp. , , -topologically sensitive) for each (3) is -topologically sensitive (resp. , , -topologically sensitive) for some

Proof. We only prove the case of -topological sensitivity since the proofs of other cases are similar: : assume that is an -sensitivity cover for and . For every , let . Set as the intersection of all the covers , i.e., is the open cover of consisting of all (nonempty) sets with the form of with for . Now, take arbitrarily an opene subset of , then . For each , there exist such that or for every . Clearly, for some and . Hence, or for every which implies that or for every . Obviously, . This turns out that is -topologically sensitive. is obvious. is easy, so we leave it to the reader.

Theorem 3. Let be a dynamical system. Then, the following statements are equivalent:(1) is -topologically sensitive (resp. , -topologically sensitive)(2) is -topologically sensitive (resp. , -topologically sensitive) for each (3) is -topologically sensitive (resp. , -topologically sensitive) for some

Proof. We only prove the case of -topological sensitivity since the proofs of other cases are similar: : assume that is an -sensitivity cover for and . For every , let . Set as the intersection of all the covers , i.e., is the open cover of consisting of all (nonempty) sets with the form of with for . Now, for any opene subset of , . For each , there exist such that or for every . Clearly, for some and . Hence, or for every which implies that or for every . Then, is obvious. is not difficult, so we leave it to the reader.

Theorem 4. If a dynamical system is topologically sensitive and there is no isolated points in and is semiopen, then is infinite for each opene subset of , where is a sensitivity cover for .

Proof. In fact, if there exists some opene set such that is finite, put , then . So, there exist such thatNote , and there exists some such that . Since is continuous, there exists such that . Without loss of generality, assume . It is clear that has a nonempty interior as which is semiopen. So, there exists an opene set and which gives that there are and such thatLet be such that and ; then,So,which implies that . That is a contradiction.

The following theorem reveals the relations between the topological sensitivity and the classical sensitivity and generalizes the result of Theorem 2.3 of [15]. At first, we recall that denotes the topology of generated by the metric of if is a metric space. Let be a family of . Review that a continuous self-map on a metric space is called -sensitivity if there exists such that, for each opene set ,

Theorem 5. Let be a continuous self-map on a metric space and let us consider the following conditions:(1) is -sensitive(2) is -topologically sensitiveThen, (1) implies (2). In addition, if is compact, then (2) is equivalent to (1).

Proof. The proof of Theorem 5 is similar to that of Theorem 2.3 in [15], so we omit it.

For proving the next theorem, we need firstly the following lemmas whose proofs are similar to those of the corresponding results of compact systems, but for the completeness, we include them in the paper.

Lemma 1. Assume that is a proper and translation invariant family of . Then, a dynamical system is -transitive if and only if it is transitive and -central.

Proof. Since is translation invariant, . If is -transitive, then it is transitive. So, is -central.
Conversely, since is transitive, for any opene subsets of , there exists such that . It is easy to prove that Noting that is -central and is translation invariant, we have , so is -transitive.

Lemma 2. A dynamical system is topologically ergodic if it is topologically transitive and the set of positive upper Banach density recurrent points of is dense in .

Proof. Assume that is an opene subset of , by the given hypothesis, . Choose ; then, has positive upper Banach density. Take with ; then, , . Set ; then, . Thus,Namely, , note which implies that , by Proposition 1, and is syndetic. Therefore, is -central. Note that is proper and translation invariant, by Lemma 1, is topologically ergodic.

Next, we give one of the main results of the paper as follows. Firstly, we review that a topological space is called a Uryshon space if for every pair of distinct points , and there are two opene sets and in such that and and . For more information about Uryshon spaces, one can refer to [18].

Theorem 6. Let be a dynamical system, where is a space. If is topologically transitive but nonminimal and the set of positive upper Banach density recurrent points of is dense in , then has an -sensitivity cover with two elements. Therefore, is -topologically sensitive.

Proof. Since is not minimal, take such that , pick , then there exist and a neighborhood of such that . Let us consider the open cover of . For any opene subset of , by Lemma 2, is syndetic. Suppose is a gap of . By the continuities of , , there exists such that for all . By Lemma 2 again, is syndetic. Suppose is a gap of . Take there exists such that , so for all . Clearly, there exists such that , so there is such that . Note that , , since , . Since is syndetic and is a translation invariant Furstenberg family, . It follows that is an -sensitivity cover for . Therefore, is -topologically sensitive.

Remark 1. Theorem 6 generalizes Theorem 2.5 of [15] to a great extent since in [17] and there exists an example showing that the set of almost periodic points of may properly be contained in .

In order to prove the next result, we firstly prove two lemmas whose proofs are same as those of the corresponding results of compact systems, but for the completeness of the paper, we present their complete proofs here.

Lemma 3. Let be a dynamical system. Then, is weakly mixing if and only if the family is a filter of , where .

Proof. If the family is a filter, then for any opene subsets of ,Especially, . So, is weakly mixing.
Conversely, suppose is weakly mixing and . By the definition of weak mixing, there exists some . Set , . For any , we haveThis means and . Then, . So, is a filter of .

Lemma 4. Let be a dynamical system. If is weakly mixing, then for each pair of opene subsets of , .

Proof. Suppose are opene subsets of . By Lemma 3, for each ,So, is thick.

Theorem 7. Suppose that is a weakly mixing and topologically ergodic dynamical system, where admits two opene subsets and such that . Then, has an -sensitivity cover with two elements. Therefore, is -topologically sensitive.

Proof. Obviously is an open cover of . Since is weakly mixing, by Lemma 3, is a filter of , where and each element of is thick. Suppose that is an opene subset of , then there exist a pair of opene subsets of such that is thick. Since is topologically ergodic, is syndetic which implies that is piecewise syndetic. Take arbitrarily and note that , then and . So, there exists such that and which yields that and . Therefore, is an -sensitivity cover for which drives that is -topologically sensitive.

4. -Topological Sensitivity of Dynamical Systems

In this section, we introduce the notion of -sensitivity for a dynamical system. One can see [19] for the same conception for compact systems and see [20] for the difference between -sensitivity and -sensitivity of compact systems.

Definition 6. Let be a dynamical system and be an open cover of and . is called an -topological sensitivity cover for if for every opene subset of , there exist and different points such that, for all with ,

Definition 7. A dynamical system (or simply the map ) is called -topologically sensitive if has an -topological sensitivity cover.

Definition 8. Let be a dynamical system, be a family of , and be an open cover of . is called an --sensitivity cover for if for each opene subset of ,

Now, by using the notion of --sensitivity cover, we introduce the topological version of -sensitivity with respect to a Furstenberg family.

Definition 9. Let be a family of . A dynamical system is called --topologically sensitive if admits an --sensitivity cover.

In the following, we present some basic properties of -topological sensitivity of dynamical systems.

Theorem 8. Let and be two dynamical systems and :(1)If and are semiconjugate and the semiconjugation from to is semi-open and is -topologically sensitive, then so is (2) is -topologically sensitive if and only if is -topologically sensitive for each if and only if is -topologically sensitive for some

Proof. (1)The proof is similar to that of (1) of Theorem 1, so we omit it(2)If is an -topological sensitivity cover for for some , then is also an -topological sensitivity cover for Now, assume that is -topologically sensitive and take an -topological sensitivity cover for . Give arbitrarily . For every , let . Set as the intersection of all the covers , i.e., is the open cover of consisting of all (nonempty) sets with the form with for . We claim that is an -topological sensitivity cover for . In fact, let be an opene subset of , then there exist and different points in such that, for any with ,Let and such that . Now, for any with ,which implies thatSo, is an -topological sensitivity cover for and is -topologically sensitive.
If is -topologically sensitive for each , then it is obvious that is -topologically sensitive for some .

About -topological sensitivity, we also have the following result that is similar to Theorem 5.

Theorem 9. Let be a continuous self-map on a metric space , and let us consider the following conditions:(1) is -sensitive(2) is -topologically sensitiveThen, (1) implies (2). In addition, if is compact, then (2) is equivalent to (1).

Proof. The proof is similar to that of Theorem 5, so we omit it.

Theorem 10. Assume that is a weakly mixing dynamical system, where is an infinite Uryshon space and there is no isolated points in . Then, is -thickly topologically sensitive for each .

Proof. Let . By the given conditions, there exist different points such that, for each , admits a neighborhood satisfying for all with . Let andWe claim that is an -thickly topological sensitivity cover for . In fact, for any opene subset of , since is weakly mixing, from Lemmas 3 and 4, it follows thatFor each , there exist distinct points such that for each ; then, for any with ,It follows that is -thickly topologically sensitive.

Remark 2. (1)Theorem 10 generalizes Proposition 2.7 of [15] to a great extent.(2)Fix a family of , one can introduce the notion of --topological sensitivity for dynamical systems by imitating the definition of -topological sensitivity. For instance, fix a family of and ; a dynamical system is called --topologically sensitive if there exists an open cover of such that, for each opene ,Then, one can obtain the similar results for --topological sensitivity as Theorems 8–10.

5. Multitopological Sensitivity of Dynamical Systems

In this section, we introduce the notion of multisensitivity for dynamical systems. Review that a compact system is multisensitive if there exists such that, for any finitely many opene subsets of , (see [2] for more details of multisensitivity of compact systems). One can refer to [10, 12, 21] for the recent developments of multisensitivity for compact systems and see [22, 23] for the new results of multisensitivity for nonautonomous systems. Now, motivated by the notion of multisensitivity of compact systems, we introduce the conception of multisensitivity for dynamical systems as follows.

Definition 10. Let be a dynamical system. An open cover of is called a multisensitivity cover for if for any finitely many opene subsets of , .

Definition 11. A dynamical system (or simply the map ) is called multisensitive if has a multisensitivity cover.

In the following, we present some basic features of multisensitivity of dynamical systems.

Theorem 11. Let and be two dynamical systems:(1)If and are semiconjugate and the semiconjugation from to is semiopen and is multisensitive, then so is (2) is multisensitive if and only if is multisensitive for each if and only if is multisensitive for some

Proof. (1)The proof is similar to that of (1) of Theorem 1, so we omit it(2)If is a multisensitivity cover for for some , then it is clear that is also a multisensitivity cover for Now, assume that is multisensitive and take a multisensitivity cover for . Give arbitrarily . For every , let . Set as the intersection of all the covers , i.e., is the open cover of consisting of all (nonempty) sets with the form of with for . We claim that is a multisensitivity cover for . In fact let be any finitely many opene subsets of , by the multisensitivity of , there exist and such thatfor each . Let and such that . Now, we have for all ,which impliesfor each . So, for every , namely, . Therefore, is a multisensitivity cover for and is multisensitive.
If is multisensitive for each , clearly is multisensitive for some .
Summarize the above proof process, and we prove result (2).

About multisensitivity, we have the following Theorem 12 that is similar to Theorems 5 and 9.

Theorem 12. Let be a continuous self-map on a metric space , and let us consider the following conditions:(1) is multisensitive(2) is multisensitiveThen, (1) implies (2). In addition, if is compact, then (2) is equivalent to (1).

Proof. The proof is similar to that of Theorem 9, so we omit it.

Theorem 13. Suppose that is a weakly mixing dynamical system and admits two opene subsets and of such that . Then, is multisensitive.

Proof. By the conditions there exist two opene sets such that . Set . Then, is a multisensitivity cover of . In fact, for any finitely many opene sets of , since is weakly mixing, by Lemma 3, and are thick for each and . Then, for each and each , there exist such that which yields that and . So, for all . It follows that . Therefore, is a multisensitivity cover for and is multisensitive.

Theorem 14. Let be an -system satisfying; there exist two minimal sets and two open sets such that , , and , then is -topologically sensitive.

Proof. We directly prove that is an -sensitivity cover for . Let be any opene subset of . For each , since are continuous and is -invariant, is an open set containing for all . By the assumption of being an -system, there is a minimal point such that for each . Let , , then is an open set containing . So, for each . Since is transitive, . Hence, there are and such that . By the continuity of , there is an open set such that and . So, there exists a minimal point of such that . Note that is also a minimal point of , then there exists a syndetic set of such that for all and each , so for each . This shows that is a thickly syndetic set. By the same argument, is also thickly syndetic. Thus, is thickly syndetic, which implies that is an -sensitivity cover for and is -topologically sensitive.

Lemma 5. If is multisensitive and there is no isolated points in , then is infinite for any finitely many opene sets of , where is a multisensitivity cover for .

Proof. In fact, if there exist finitely many opene sets of such that is finite, putthen for all . Thus, there exist such thatfor every . Note that, for every , there exists such that . By the continuities of , , there is an open neighbourhood of such that for each and . Now, for , let , and it is easy to check that for every and . Thus, and . That is a contradiction.

Theorem 15. Let be a dynamical system, where is a surjection. If is multisensitive, then is thickly topologically sensitive. Moreover, if is point-transitive, then the converse holds.

Proof. Firstly assume that is multisensitive with a multisensitivity cover of and . For any opene set , we choose opene sets for each . By Lemma 5, is infinite, so we can choose some with . Note that for each , and there exist which gives that such thatThen, for each , there exist such that , andwhich implies for all . Therefore, is thickly topologically sensitive.
Now, we assume is thickly topologically sensitive with a thickly topological sensitivity cover of and is a transitive point of . Let and be opene subsets of . Then, for each , there exists such that . Hence, there exists such that for each . By the given assumptions, there is such thatSo, for all which yields that there exist such thatfor all . Therefore, for each andwhich implies that for every , i.e., . Therefore, is multisensitive.

Remark 3. For the related studies of thickly topological sensitivity and multisensitivity of semigroups, one can refer to [24] in which all kinds of sensitivities are defined via a uniform structure of the involved space. In this paper, we introduce such sensitivities by an open cover of the phase space, so they are a little different.

Data Availability

No data are used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11261039 and 11661054).

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Copyright © 2020 Tengfei Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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