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Discrete Dynamics in Nature and Society
Volume 2012, Article ID 970934, 7 pages
http://dx.doi.org/10.1155/2012/970934
Research Article

Uniform Convergence and Transitive Subsets

1School of Mathematics, Sichuan University, Sichuan, Chengdu 610064, China
2Department of Mathematics, Shangqiu Normal University, Henan, Shangqiu 476000, China

Received 26 September 2011; Accepted 9 December 2011

Academic Editor: Recai Kilic

Copyright © 2012 Lei Liu 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.

Abstract

Let be a metric space and a sequence of continuous maps that converges uniformly to a map . We investigate the transitive subsets of whether they can be inherited by or not. We give sufficient conditions such that the limit map has a transitive subset. In particular, we show the transitive subsets of that can be inherited by if converges uniformly strongly to .

1. Introduction

A topological dynamical system 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. Define by the set of all positive integers.

In [1], Blanchard and Huang introduced the concepts of weakly mixing subset and partial weak mixing, derived from a result given by Xiong and Yang [2] and showed “partial weak mixing implies Li-Yorke chaos” and “Li-Yorke chaos does not imply partial weak mixing”. A closed set 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 a such that for . A topological dynamical system is called partial weak mixing if contains a weakly mixing subset. Motivated by the idea of Blanchard and Huang’s notion of “weakly mixing subset”, Oprocha and Zhang [3] extended the notion of weakly mixing subset and gave the concept of “transitive subset” and discussed its basic properties.

It is a well-known fact that if a sequence of continuous maps converges uniformly, then the uniform limit map is continuous. Abu-Saris and Al-Hami [4] studied uniform convergence and chaotic behavior. Later Abu-Saris et al. [5] pointed out some wrong claims in [4] and corrected them. Román-Flores [6] gave sufficient conditions for the topological transitivity of uniform limit map of a sequence of continuous maps , where is a compact metric space. Fedeli and Le Donne [7] studied the dynamical behavior of the uniform limit of a sequence of continuous self-maps on a compact metric space satisfying topological transitivity or other related properties and gave some conditions for the transitivity of a limit. Bhaumik and Choudhury [8] investigated the chaotic behavior of the uniform limit map of a sequence of continuous topologically transitive maps , where is a compact interval. Recently, Yan, Zeng, and Zhang et al. [9] studied transitivity and sensitive dependence on initial conditions for uniform limits.

In this paper, motivated by the idea of Román-Flores [6], we give sufficient conditions such that the limit map has a transitive subset. In particular, we prove that is a transitive subset of if is a transitive subset of for every when a sequence of continuous maps converges strongly uniformly to a map , where is a compact metric space. Moreover, we give an example to show that if is a transitive subset of , then cannot be a transitive subset of for some .

2. Preliminaries

Topological transitivity (see [1012]) are global characteristic of topological dynamical systems. Let be a topological dynamical system. is topologically transitive if for any nonempty open subsets and of there exists a such that . For a topological dynamical system , the orbit of is the set for every . is point transitive if there exists a point with dense orbit, that is, . Such a point is called a transitive point of . By [13], if is a compact metric space without isolated points, then the topologically transitive and point transitive are equivalent.

Definition 2.1 (see[3]). A closed subset is called a transitive subset of if for any choice of nonempty open subset of and nonempty open subset of with , there exists a such that .

Remark 2.2. (1) By Definition 2.1, is transitive if and only if is a transitive subset of .
(2) If is a transitive point of , then is a transitive subset of .

Definition 2.3 (see[14]). Let be a topological space. and are two nonempty subsets of . is dense in if .

In fact, we easily prove that is dense in if and only if for any nonempty open set of .

Proposition 2.4. Let be a topological dynamical system and be a nonempty closed set of . Then the following conditions are equivalent.(1) is a transitive subset of . (2)Let be a nonempty open subset of and a nonempty open subset of with . Then there exists such that . (3)Let be a nonempty open set of with . Then is dense in .

Proof. Let be a transitive subset of . Then for any choice of nonempty open set of and nonempty open set of with , there exists such that . Since , it follows that .
Let be a nonempty open set of and be a nonempty open set of with . By the assumption of (2), there exists such that . Furthermore,
Hence, is dense in .
Let be a nonempty open set of and a nonempty open set of with . Since is dense in , it follows that . Hence, there exists such that . Moreover, , which implies . Therefore, is a transitive subset of .

Definition 2.5. Let be a metric space and a sequence of continuous maps , for each . is said to converge strongly uniformly to if for any , there exists such that for any , and satisfying If converges strongly uniformly to , is called a strong uniform convergent sequence.

The following example is from [9, 15]; we show that the example is a strong uniformly convergence example.

Example 2.6. Let . Denote for any and . Let satisfy For any , we define satisfying Then it is easy to see that is a continuous map for each and converges strongly uniformly to , the identity on .

3. Main Results

Let denote the set of continuous maps . In the sequel, as in usual, denotes the uniform metric on , that is, . A topological space is perfect if is closed and has no isolated points. Clearly, if is a perfect space, then any nonempty open set of has no isolated points.

From the idea of Román-Flores [6], we obtain the following theorem.

Theorem 3.1. Let be a compact metric space and a sequence of continuous maps that converges uniformly to a map . Assume that is a perfect set of and is a transitive subset of for all . Additionally, suppose that
(1) as ,
(2) is dense in , for some .
Then is a transitive subset of .

Proof. Let be a nonempty open set of and a nonempty open set of with . Since condition (2), there exists such that is dense in . Furthermore, by condition (1) and is perfect, we obtain that the sequence is also dense in . Moreover, is a nonempty open set of ; there exists such that . Let . Then is a nonempty open set of . Since is a perfect metric space and is dense in , there exists such that . Hence, we have Consequently, . Therefore, is a transitive subset of .

Theorem 3.2. Let be a compact metric space. Assume a sequence of continuous maps that converges strongly uniformly to a map and is a transitive subset of dynamical systems for each . Then is a transitive subset of .

Proof. Let be a nonempty open set of and a nonempty open set of with . Since is a compact metric space and , there exists a nonempty open set of such that and .
Let for each . Since is a transitive subset of for each , by Proposition 2.4, then is an open set of and is dense in . We denote . By Baire theorem, is dense in . Furthermore, we have . Take a point . There exists such that for each . Denote for each . Without loss of generality, we may assume because is a compact metric space. Choose a such that . Since maps sequence converges strongly uniformly to and , there exists such that
It follows that , which implies . Therefore, . This shows that is a transitive subset of .

The following example is from [4]. We give the example which shows if maps sequence converges uniformly to a map and is a transitive subset of for each , then cannot be a transitive subset of .

Example 3.3 (see [4]). Let be the unit circle and a translation map such that Let be an irrational number, , and such that . Let maps sequence converge uniformly to a map . Then is not topologically transitive on ; that is, is not a transitive subset of dynamical system .
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 [16], if is an irrational number, then is topologically transitive on . Therefore, is a transitive subset of . Since is an irrational number for each , then is topologically transitive for each , which implies is a transitive subset of for each . Moreover, maps sequence converges uniformly to a map , where is identity map. Therefore, is not topologically transitive on , which implies is not a transitive subset of .
Let be a continuous map for each , and maps sequence converges uniformly to a map . The following example shows that is a transitive subset of , but there exists such that is not a transitive subset of .

Example 3.4. Let Observe that the given sequence converges uniformly to tent map Figures 1 and 2, which is known to be topologically transitive on (see [16]). We will prove that is a transitive subset of .
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 exists and such that . Furthermore, we have . Thus, for any nonempty open set of and nonempty open set of with , there exists such that . This shows that is a transitive subset of . Moreover, and for all , which implies that is not a transitive subset of .

970934.fig.001
Figure 1
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Figure 2

Acknowledgments

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 Natural Science Foundation of Henan Province (092300410148), China.

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