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Journal of Applied Mathematics
Volume 2011 (2011), Article ID 838639, 9 pages
http://dx.doi.org/10.1155/2011/838639
Research Article

3-Adic System and Chaos

1School of Science, Dalian Nationalities University, Liaoning, Dalian 116600, China
2School of Information and Computing Science, Beifang University of Nationality, Ningxia, Yinchuan 750021, China
3Department of Mathematics, Liaoning Normal University, Liaoning, Dalian 116029, China
4Institute of Applied Physics and Computational Mathematics, Beijing 100094, China

Received 1 August 2011; Accepted 3 October 2011

Academic Editor: James Buchanan

Copyright © 2011 Lidong 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.

Abstract

Let be a 3-adic system. we prove in the existence of uncountable distributional chaotic set of , which is an almost periodic points set, and further come to a conclusion that is chaotic in the sense of Devaney and Wiggins.

1. Introduction

In 1975, Li and Yorke introduced in [1] a new definition of chaos for interval maps. The central point in their definition is the existence of a scrambled set. Later, it was observed that positive topological entropy of interval map implies the existence of a scrambled set [2]. Many sharpened results come into being in succession (see [311]). One can find in [3, 4, 12] equivalent conditions for to be chaotic and in [13] or [14] a chaotic map with topological entropy zero, which showed that positive topological entropy and Li-Yorke chaos are not equivalent.

By the result, it became clear that the positive topological entropy is a much stronger notion than the definition of chaos in the sense of Li and Yorke. To remove this disadvantage, Zhou [15] introduced the notion of measure center and showed importantly dynamical properties of system on its measure center. To decide the concept of measure center, he defined weakly almost periodic point, too, showing that the closure of a set of weakly almost periodic points equals to its measure center and the set of weakly almost periodic points is a set of absolutely ergodic measure 1. These show that it is more significant to discuss problems on a set of weakly almost periodic points. On the other hand, one important extensions of Li-Yorke definition were developed by Schweizer and Smítal in [16]; this paper introduced the definition of distributional chaos and prove that this notion is equivalent to positive topological entropy for interval maps. And many scholars (such as Liao, Du, and Zhou, Wang) proved that the positive topological entropy of interval map is equivalent to the uncountable Li-Yorke chaotic set and the uncountable distributional chaotic set for , , and . Meanwhile Liao showed that the equivalent characterization is no longer valid when acts on more general compact metric spaces.

In this paper, we discuss the existence of uncountable distributional chaotic set of in 3-adic system.

The main results are stated as follows.

Main Theorem 1. Let be a 3-adic system. Then(1) contains an uncountable distributional chaotic set of ;(2) is chaotic in the sense of Devaney;(3) is chaotic in the sense of Wiggins.

2. Basic Definitions and Preparations

Throughout this paper, will denote a compact metric space with metric , is the closed interval [0, 1].

For a continuous map : , we denote the set of almost periodic points of by and denote the topological entropy of by , whose definitions are as usual; will denote the -fold iteration of .

For , in , any real number and positive integer , let

where we use to denote the cardinality of a set. Let

Definition 2.1. Call , a pair of points displaying distributional chaos, if(1) for some ;(2) for any .

Definition 2.2. is said to display distributional chaos, if there exists an uncountable set such that any two different points in display distributional chaos.

Definition 2.3. Let be a metric space and be a continuous map. The dynamical system is called chaotic in the sense of Devaney, if(1) is transitive;(2)the periodic points are dense in ;(3) is sensitive to initial conditions.

Definition 2.4. Let be a metric space and be a continuous map. The dynamical system is called chaotic in the sense of Wiggins, if there exists a compact invariant subset such that(1) is sensitive to initial conditions;(2) is transitive.

Definition 2.5. Let and be dynamical systems; if there exists a homeomorphism such that , then and are said to be topologically conjugate.

The notion of adic system is defined as follows.

Definition 2.6. Put We use the sequence to denote simply the member in . Define as follows: for any , if , , then It is not difficult to check that is a metric on and is a compact abelian group. Define by for ; or is called the 3-adic system. (see [17])
Call an invariant closed set   3-adic, if the restriction is topologically conjugate to the 3-adic system.
Consider the following functional equation: where (0,1) is to be determined, and is the 3-fold iteration of .
By we denote the set of continuous solutions of (2.5) such that any satisfies: there exists such that ; the restrictions and are both once continuously differentiable, and on , on ; .

The following Lemma can be concluded by in [18, Theorem  2.1].

Lemma 2.7. Let 0 . Let be on each of the interval and , and satisfy(1);(2) on and on ;(3)there exists such that and ;(4). Then there exists a unique with . Conversely, if is the restriction on of some , then it must satisfy (1)–(4).

Proposition 2.8. .

Proof. Let , . Define by
It is not difficult to check that satisfies the condition (1)–(4) in Lemma 2.7. So .
We will be concerned in the notions of Hausdorff metric and Hausdorff dimension, whose definitions can be found in [19].

Lemma 2.9 (see [19, Theorem 8.3]). Let be contractions on . Then there exists a unique nonempty compact set such that where is a transformation of subsets of . Furthermore, for any nonempty compact subset of , the iterates converge to in the Hausdorff metric as .

Lemma 2.10 (see [19, Theorem 8.8]). Let be contractions on R for which the open set condition holds; that is, there is an open interval such that(1), (2) are pairwise disjoint.Moreover, suppose that for each , there exists , such that for all . Then , where denotes the Hausdorff dimension and is defined by

Lemma 2.11 (see [20, Theorem 3.2], [21]). Let be continuous. Then the followings are equivalent:(1);(2) contains an uncountable distributional chaotic set of .

Lemma 2.12 (see [21]). Let , be continuous, where , are compact metric spaces. If there exists a continuous surjection such that , then .

Lemma 2.13 (see [22]). Let and be recurrent but not periodic such that . Then the inequality holds for all even m and all odd .

Lemma 2.14 (see [23, Theorem ]). Let be an interval map. Then if and only if there exists a closed invariant subset such that is chaotic in the sense of Devaney.

Lemma 2.15 (see [23, Theorem ]). Let be an interval map. If , then is chaotic in the sense of Wiggins.

3. Proof of Main Theorem

In the sequel, we always suppose that and take the minimum at .

Let , . For , define by , , . Then is a contraction for every . Let . By Lemma 2.9, there exists a unique nonempty compact set with For simplicity, we write for .

Step 1. Prove that for any , , , .

Proof. Letting act on both sides of the equality , we get immediately the first equality. A similar argument yields the second equality. To show the third equality, we write (2.5) as . Since , it follows from Lemma 2.7 that and . By this and definitions of and , we get

Step 2. Prove that for any subsets and , there is an such that .

Proof. If , , then by Step 1. Using this repeatedly, we get for any If for each , we all have , then from (3.3), (nothing that ). Thus the lemma holds for this special case. Assume that there exists some , , such that for , but . Then by using (3.3) repeatedly, we know that or has the form , where for . Continuing this procedure, we must get some , such that .
In, Steps 3, 5, and 6, we always suppose that the notation is as in (3.1).

Step 3. Prove that

Proof. Since , we have for any . So from Lemma 2.9 we get

Step 4. Prove that for any , is an invariant set of , that is, .

Proof. Note that each has the form or or . Then, by using Step 1 repeatedly, we have Thus by , we have . Moreover,

Step 5. Prove that the restriction is topologically conjugate to , where is the 3-adic system as defined in Section 1.

Proof. By the definition of , we have with this union disjoint. Then transforming by , again with a disjoint union. Thus the sets (with arbitrary) form a net in the sense that any pair of sets from the collection are either disjoint or such that one is included in the other. It follows from Step 3 that for any , if let then is nonempty, and if , then there exists a unique with .
We now define a map of onto by setting if . Then is well defined. It is easy to see that for each , the contraction ratio of , so the contraction ratio of . It follows that diam converges to zero uniformly for as (where diam denotes diameter). Thus is a single point for each . And so is injective. Moreover the map is continuous. Let be the least distance between any two of the interval . If , , and , then . Finally, since by (3.7), we have for each .

Step 6. Prove that if has an -adic set and the is not a power of 2, then .

Proof. Write , where is odd and is an integer. Let be the -adic set of and . There exists a homeomorphism such that for , . We may assume without loss of generality that . Put Then is an open neighborhood of the sequence . There exists an , such that for any , if , there . Note that for and furthermore , we have that there exists an such that Let . Since we easily see that if and only if divides , it follows that , since can not divide . And so . In particular, . By the same argument, we also have . In particular, . Since , from (3.12), , that is, . Thus we have for the odd , Note that is current and nonperiodic for , and so is for . By Lemma 2.13 we get . Moreover .
Finally, we prove that contains an uncountable distributional chaotic set of . By Step 5, the restriction is topologically conjugate to . Thus there is a homeomorphism such that for any ,
According to Lemma 2.11, there is an uncountable set , which is distributional chaotic. By Lemma 2.12 for any , there exists such that . Let Then is an uncountable set.
To complete the proof, it suffices to show that is a distributional chaotic set for .
First of all, we prove that for any , if for some , then for some .
For given  , by uniform continuity of , there exists , such that for any , , provided . Since we easily see that , it follows that if , then This implies for any . Thus by the definition of , we immediately have the following result:
Secondly, we prove that if for all , then for all . Since is homeomorphism, is a surjective continuous map. By the first proof, we have which gives By (3.18), (3.20), and the arbitrariness of and , we conclude that is an uncountable distributional chaotic set of .
The proofs of (2) and (3) of the Main Theorem are obvious.

Acknowledgments

This work is supported by the major basic research fund of Department of Education of Liaoning Province no. 2009A141 and the NSFC no. 10971245 and the independent fund of central universities no. 10010101.

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