Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 281395, 6 pages

http://dx.doi.org/10.1155/2013/281395

Research Article

On Local Aspects of Topological Transitivity and Weak Mixing in Set-Valued Discrete Systems

Lei Liu

Department of Mathematics, Shangqiu Normal University, Shangqiu, Henan 476000, China

Received 3 March 2013; Accepted 15 May 2013

Academic Editor: Carlo Piccardi

Copyright © 2013 Lei Liu. 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.

Abstract.

Blanchard and Huang introduced the notion of weakly mixing subset, and Oprocha and Zhang gave the concept of transitive subset and studied its basic properties. In this paper our main goal is to discuss the weakly mixing subsets and transitive subsets in set-valued discrete systems. We prove that a set-valued discrete system has a transitive subset if and only if original system has a weakly mixing subset. Moreover, we give an example showing that original system has a transitive subset, which does not imply set-valued discrete system has a transitive subset.

1. Introduction

Throughout this paper a topological dynamical system (abbreviated to TDS) is a pair , where is a compact metric space with metric and is a continuous map. When is finite, it is a discrete space and there is no any nontrivial convergence. Hence, we assume that contains infinitely many points. Let denote the set of all positive integers and let .

Topological transitivity, weak mixing, and sensitive dependence on initial conditions (see [1–4]) are global characteristics of topological dynamical systems. Let be a TDS. is (topologically) transitive if for any nonempty open subsets and of there exists an such that . is (topologically) weakly mixing if for any nonempty open subsets , and of , there exists an such that and . It follows from these definitions that weak mixing implies transitivity.

In [5], Blanchard introduced overall properties and partial properties. 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 [1, 8]) belong to partial properties. Weak mixing is an overall property; it is stable under semiconjugate maps and implies Li-Yorke chaos. We have a weakly mixing system that always contains a dense uncountable scrambled set (see [9]). In [10], Blanchard and Huang introduced the concepts of weakly mixing subset, derived from a result given by Xiong and Yang [11] and showed “partial weak mixing implies Li-Yorke chaos” and “Li-Yorke chaos can not imply partial weak mixing.”

Motivated by the idea of Blanchard and Huang's notion of “weakly mixing subset,” Oprocha and Zhang [12] extended the notion of weakly mixing subset, gave the concept of “transitive subset,” and discussed its basic properties. In recent years, many authors studied the dynamical properties for set-valued discrete systems. Román-Flores [13], Banks [14], Peris [15], Wang and Wei [16], and Acosta et al. [17] investigated the properties of topological transitivity and weak mixing for set-valued discrete systems. Fedeli [18], Guirao et al. [19] and Hou et al. [20] studied Devaney chaos for set-valued discrete systems. Lampart and Raith [21] discussed topological entropy for set-valued maps. Liu et al. [22] and Wang et al. [23] studied sensitivity of set-valued discrete systems. Wu and Xue [24] discussed shadowing property for induced set-valued dynamical systems. Also, we continue to discuss transitive subsets, weakly mixing subsets for set-valued discrete systems, and investigate the relationship between set-valued discrete system and original system on transitive subset, weakly mixing subset. More precisely, a set-valued discrete system has a transitive subset if and only if original system has a weakly mixing subset and we give an example showing that original system has a transitive subset which does not imply set-valued discrete system has a transitive subset. Moreover, we prove that a transitive point of set-valued discrete system is a transitive subset of original system.

2. Preliminaries

A TDS is point transitive if there exists a point with dense orbit, that is, , where denotes the closure of . Such a point is called transitive point of . If is a compact metric space without isolated points, then topologically transitive and point transitive are equivalent (see [2]). A TDS is minimal if for every ; that is, every point is transitive point. A point is called minimal if the subsystem is minimal.

The distance from a point to a nonempty set in is defined by

Let be the family of all nonempty compact subsets of . The Hausdorff metric on is defined by

It follows from Michael [25] and Engelking [26] that is a compact metric space. The Vietoris topology on is generated by the base

where are open subsets of .

Let be the induced set-valued map defined by

Then is well defined. is called a set-valued discrete system.

Let be space; that is, single point set is closed. Then is a closed subset of for any nonempty closed subset of (see [25]).

Definition 1 (see [10]). Let be a TDS and let be a closed subset of with at least two elements. is said to be weakly mixing if for any , any choice of nonempty open subsets of and nonempty open subsets of with , , there exists an such that for . is called partial weak mixing if contains a weakly mixing subset.

Definition 2 (see [12]). Let be a TDS and be a nonempty subset of . is called a transitive subset of if for any choice of nonempty open subset of and nonempty open subset of with , there exists an such that .

Remark 3. (1) is topologically transitive if and only if is a transitive subset of .

(2) By [12], is a transitive subset if and only if is a transitive subset, where denotes the closure of .

According to the definitions of transitive subset and weakly mixing subset, we have the following.

Result 1. If is a weakly mixing subset of , then is a transitive subset of .

Result 2. If is a transitive point of , then is a transitive subset of .

Result 3. If is a periodic orbit of for some , then is a transitive subset of .

Example 4. Tent map

is shown in Figures 1 and 2, which is known to be transitive on (see [6]). We prove that is a transitive subset of .

Figure 1

Figure 2

Let denote the set of extreme value points of for every . Then  . Since , , , and , we have

Let for . Then . For any nonempty open set of , without loss of generality, we take for a given and , where denotes the interior of . When and , then there exist and such that . Furthermore, we have . Thus, for any nonempty open set of and nonempty open set of with , there exists a such that . This shows that is a transitive subset of .

Definition 5 (see [27]). Let be a topological space and be a nonempty set of . is a regular closed set of if , where denotes the interior of .

We easily prove that is a regular closed set if and only if for any nonempty set of .

Theorem 6 (see [14, 15]). Let be a compact space, and let be equipped with the Vietoris topology. If is a continuous map, then is continuous and is weakly mixing is weakly mixing is topologically transitive.

3. Transitive Subsets and Weakly Mixing Subsets of Set-Valued Discrete Systems

For a TDS and two nonempty subsets , we use the following notation:

Theorem 7. is a weakly mixing subset of if and only if is a weakly mixing subset of .

Proof

Necessity. We prove for any , any choice of nonempty open subsets of and nonempty open subsets of with for , that there exists an such that

For nonempty open subset of , there exist open subsets of such that for . Without loss of generality, let

and are nonempty open subsets of for . Furthermore,

Let . Then we have for . Moreover, , then for .

We consider any nonempty open subsets of and any nonempty open subsets of with for , . Since is a weakly mixing subset of , then there exists an such that

Take for , . We have and for , . Let . Then and . Furthermore, we have . Therefore, for .

Sufficiency. We show that for any , any choice of nonempty open subsets of and nonempty open subsets of with for each , there exists an such that for .

For nonempty open subset for , there exists an open subset of such that for . Let

Then is a nonempty open subset of and is a nonempty open set of with for . Since is a weakly mixing subset of , there exists an such that

Take . We have and for . Therefore, . Furthermore, we have for . This shows is a weakly mixing subset of .

Theorem 8. Let be a nonempty closed set of . If is a transitive subset of , then is a transitive subset of .

Proof. We show that for any choice of nonempty open subset of and nonempty open subset of with , there exists an such that .

For nonempty open subset of , there exists a nonempty open subset of such that . Let , , and ; then is a nonempty open subset of . Moreover, implies that . Since is a topologically transitive subset of , there exists an such that . Furthermore, there exists such that and , which implies and . Therefore, we have .

Lemma 9. Let be a regular closed set of but not a singleton. is a weakly mixing subset of if and only if for any choice of nonempty open subsets of and nonempty open subsets of with , , there exists an such that for .

Proof. Necessity is obvious by the definition of weakly mixing subset. We need only to prove sufficiency.

Let be two nonempty open subsets of and let be two nonempty open subsets of with , . Since is a regular closed set of , then . We consider two nonempty open subsets , of and two nonempty open subsets , of ; there exists an such that and . Furthermore, we have and .

Let and . Then is a nonempty open subset of and is a nonempty open subset of with . By assumption, . For any , we have , which implies . Since , , it follows that

Hence, . Furthermore, we have and  . This shows that for any , any choice of nonempty open subsets of and nonempty open subsets of with for , we have . This means that there exists an such that for . Therefore, is a weakly mixing subset of .

Lemma 10. Let be a regular closed set of but not a singleton. is a weakly mixing subset of if and only if for any choice of nonempty open subset of and nonempty open subsets of with and , there exists an such that and .

Proof. Necessity is obviously by the definition of weakly mixing subset. We need only prove sufficiency.

By Lemma 9, we only prove that for any choice of nonempty open subsets of and nonempty open subsets of with , , there exists an such that for .

Let and be two open sets of satisfying and . Since is a regular closed set, then and . We consider nonempty open subset of and nonempty open subsets , of , according to the assumption that there exists an such that

Moreover, and are nonempty open sets of with and . We consider nonempty open subset of and nonempty open subsets , of ; there exists an such that and . As  , we have . Since , then , which implies . Therefore, by Lemma 9, is a weakly mixing subset of .

Theorem 11. Let be a regular closed subset of but not a singleton. If is a transitive subset of , then is a weakly mixing subset of .

Proof. Suppose is a regular closed subset of but not a singleton. Then is a closed subset of but not a singleton. Let is a nonempty open subset of , and let and be two nonempty open subsets of with and . According to Lemma 10, we only prove there exists an such that and .

For nonempty open subset of , there exists an open subset of such that . Let and , then and are open subsets of with and . We consider nonempty open subset of and nonempty open subset of . Since is a transitive subset of , there exists an such that

Take . We have , , and , that is, , , and , which implies and . This shows is a weakly mixing subset of .

By Theorems 7 and 11, we have the following corollary.

Corollary 12. Let be a regular closed subset of but not a singleton. Then the following properties are equivalent:

(1)is a weakly mixing subset of ; (2) is a weakly mixing subset of ; (3) is a transitive subset of .

Lemma 13. Let be a transitive point of . Then is a transitive point of for every .

Proof. Suppose that is a transitive point of . Then for any open set of , there exists such that

In particular, take ; there exists such that . Furthermore, for any , we have . Since is any nonempty open set of , it follows that is a transitive point of .

Theorem 14. Let be a transitive point of . Then is a transitive subset of .

Proof. Suppose that is a transitive point of . Then is a nonempty closed set of . Let be a nonempty open set of and let be a nonempty open set of with . We prove that there exists an such that .

Since is a transitive point of , by Lemma 13, is a transitive point of for every . Let , where is an open set of . Let . Then . It means that there exists an such that . Furthermore, we have . Therefore, is a transitive subset of .

Example 15. Let be the unit circle and let be a translation map such that

If is an irrational number, then is a transitive subset of , but is not a transitive subset of .

It is well known that if is a rational number, then all points are periodic of period , and so the set of periodic points is, obviously, dense in . Moreover, by Jacobi’s Theorem [6], if is an irrational number, then each orbit is dense in . Since is a compact metric space. Hence, is topologically transitive.

Let . Then is a nonempty closed subset of . Let be a nonempty subset of and is an open subset of with . Take . Since is dense in , there exists an such that . Furthermore, we have . Therefore, is a transitive subset of ; that is, is a transitive subset of .

Let . Then , where denotes the diameter of . Put such that . Let and . Then is a nonempty open subset of and is an open subset of with . Moreover, for any and any , we have and . Furthermore, for all . Therefore, for all . It means that is not a transitive subset of .

Example 16. Let . Define by

Then has a weakly mixing subset (Figures 3 and 4).

Figure 3

Figure 4

Let and . Then and . Hence, is equal to the tent map of Example 4. Furthermore, by [8], is mixing. Hence, is a weakly mixing subset of . We prove that is a weakly mixing subset of .

For any , any choice of nonempty open subsets of and nonempty open subsets of with , , we have are nonempty open subsets of for all . Since is weak mixing, by [28], there exists an such that for . Furthermore, we have for . Hence, is a weak mixing subset of . By Theorem 7, is a weakly mixing subset of .

Acknowledgments

The author would like to thank the referees for many valuable and constructive comments and suggestions for improving this paper. This work is supported by the Natural Science Foundation of Henan Province (122300410427), China.

References

L. S. Block and W. A. Coppel, Dynamics in One Dimension, Springer, Berlin, Germany, 1992.

S. Kolyada and L. Snoha, “Some aspects of topological transitivity-a survey,” Grazer Mathematische Berichte, vol. 334, pp. 3–35, 1997.

C. Robinson, Dynamical Systems, CRC Press, Boca Raton, Fla, USA, 1999.

P. Walters, An Introduction to Ergodic Theory, vol. 79 of Graduate Texts in Mathematics, Springer-Verlag, New York, NY, USA, 1982.

F. Blanchard, “Topological chaos: what may this mean?” Journal of Difference Equations and Applications, vol. 15, no. 1, pp. 23–46, 2009.

R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, Redwood City, Calif, USA, 2nd edition, 1989.

T. Y. Li and J. A. Yorke, “Period three implies chaos,” The American Mathematical Monthly, vol. 82, no. 10, pp. 985–992, 1975.

S. Ruette, “Chaos for continuous interval maps: a survey of relationship between the various sorts of chaos,” http://www.math.u-psud.fr/~ruette/

A. Iwanik, “Independence and scrambled sets for chaotic mappings,” in The Mathematical Heritage of C.F. Gauss, pp. 372–378, World Scientific, River Edge, NJ, USA, 1991.

F. Blanchard and W. Huang, “Entropy sets, weakly mixing sets and entropy capacity,” Discrete and Continuous Dynamical Systems A, vol. 20, no. 2, pp. 275–311, 2008.

J. C. Xiong and Z. G. Yang, “Chaos caused by a topologically mixing map,” in Dynamical Systems and Related Topics, vol. 9, pp. 550–572, World Scientific, Singapore, 1991.

P. Oprocha and G. Zhang, “On local aspects of topological weak mixing in dimension one and beyond,” Studia Mathematica, vol. 202, no. 3, pp. 261–288, 2011.

H. Román-Flores, “A note on transitivity in set-valued discrete systems,” Chaos, Solitons and Fractals, vol. 17, no. 1, pp. 99–104, 2003.

J. Banks, “Chaos for induced hyperspace maps,” Chaos, Solitons and Fractals, vol. 25, no. 3, pp. 681–685, 2005.

A. Peris, “Set-valued discrete chaos,” Chaos, Solitons and Fractals, vol. 26, no. 1, pp. 19–23, 2005.

Y. Wang and G. Wei, “Characterizing mixing, weak mixing and transitivity of induced hyperspace dynamical systems,” Topology and its Applications, vol. 155, no. 1, pp. 56–68, 2007.

G. Acosta, A. Illanes, and H. Méndez-Lango, “The transitivity of induced maps,” Topology and Its Applications, vol. 156, no. 5, pp. 1013–1033, 2009.

A. Fedeli, “On chaotic set-valued discrete dynamical systems,” Chaos, Solitons and Fractals, vol. 23, no. 4, pp. 1381–1384, 2005.

J. L. G. Guirao, D. Kwietniak, M. Lampart, P. Oprocha, and A. Peris, “Chaos on hyperspaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 1-2, pp. 1–8, 2009.

B. Hou, X. Ma, and G. Liao, “Difference between Devaney chaos associated with two systems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1616–1620, 2010.

M. Lampart and P. Raith, “Topological entropy for set valued maps,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 6, pp. 1533–1537, 2010.

H. Liu, E. Shi, and G. Liao, “Sensitivity of set-valued discrete systems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 82, pp. 985–992, 2009.

Y. Wang, G. Wei, and W. H. Campbell, “Sensitive dependence on initial conditions between dynamical systems and their induced hyperspace dynamical systems,” Topology and its Applications, vol. 156, no. 4, pp. 803–811, 2009.

Y. Wu and X. Xue, “Shadowing property for induced set-valued dynamical systems of some expansive maps,” Dynamic Systems and Applications, vol. 19, no. 3-4, pp. 405–414, 2010.

E. Michael, “Topologies on spaces of subsets,” Transactions of the American Mathematical Society, vol. 71, pp. 152–182, 1951.

R. Engelking, General Topology, PWN, Warsaw, Poland, 1977.

R. F. Dickman Jr., “Regular closed maps,” Proceedings of the American Mathematical Society, vol. 39, pp. 414–416, 1973.

H. Furstenberg, “Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation,” Mathematical Systems Theory, vol. 1, pp. 1–49, 1967.