Difference between Devaney chaos associated with two systems

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Abstract

We discuss the relation between Devaney chaos in the base system and Devaney chaos in its induced hyperspace system. We show that the latter need not implies the former. We also argue that this implication is not true even in the strengthened condition. Additionally we give an equivalent condition for periodic density in the hyperspace system.

Introduction

Let (X,d) be a compact metric space with metric d and f:XX be continuous. Then (X,f) is called a compact system. For every positive integer n, we define fn inductively by fn=ffn1, where f0 is the identity map on X.

A compact system (X,f) is called Devaney chaotic [1] if it satisfies the following three conditions:

(1) f is transitive, i.e., for every pair U and V of non-empty open subsets of X there is a non-negative integer k such that fn(U)V;

(2) f is periodically dense, i.e., the set of periodic points of f is dense in X;

(3) f is sensitive, i.e., there is a δ>0 such that, for any xX and any neighborhood V of x, there is a non-negative integer n such that d(fn(x),fn(y))>δ.

It is worth noting that the conditions (1) and (2) imply that f is sensitive if X is infinite [2], [3]. This means that the condition (3) is redundant in the above definition.

In the paper [4] Román-Flores investigated a certain hyperspace system (K(X),f¯) associated to the base system (X,f), where f¯:K(X)K(X) is the natural extension of f and K(X) is the family of all non-empty compact sets of (X,d) endowed with the Hausdorff metric induced by d. He presented a fundamental question: Does the chaoticity of (X,f) (individual chaos) imply that of (K(X),f¯) (collective chaos)? and conversely?

As a partial response to this question, Román-Flores [4] discussed the transitivity of the two systems, and showed that the transitivity of f¯ implies that of f, but the converse is not true. Fedeli [5] showed that the periodical density of f implies that of f¯. Banks gave an example which has a dense set of periodic points in the hyperspace system but has none in the base system [3]. Gongfu Liao showed that there is an example on the interval which is Devaney chaos in the base system while is not in its induced hyperspace system [6].

In this paper, we show that f¯ being Devaney chaos need not imply f being Devaney chaos. Further, f need not be Devaney chaos even if f¯ is mixing and periodically dense. This answers the question posed by Román-Flores and Banks [3], [12].

Section snippets

Preliminaries

(X,f) is a compact system. If YX and Y is f-invariant, i.e. f(Y)Y, then (Y,f|Y) is called the subsystem of (X,f) or f, where f|Y is the restriction of f on Y.

A subset AX is minimal if the closure of the orbit of any point x of A is A, i.e. Orb(x)¯=A, for all xA. An equivalent notion is that A has no proper f-invariant closed subset.

A point xX is said to be an almost periodic point if for any ε>0 exists NN such that for any integer q1 exists an integer r with property that qr<N+q and fr(x

Periodically dense system

Theorem 5

If (K(X),f¯) is the hyperspace system induced by the compact system (X,d) , then K(X) has a dense set of periodic points if and only if for any non-empty open subset U of Xthere exists a compact subset KU and an integer n>0 such that fn(K)=K .

Proof

Since K(X) has a dense set of periodic points, every non-empty open subset of K(X) has at least one periodic point. Further, every basic element has at least one periodic point. Let U be an arbitrary open subset of X. Then by the definition of the

Devaney chaos

If a is an almost periodic point of Σ2 and X=orb(a)¯, then (X,σ) is usually a mixing non-trivial minimal subsystem of (Σ2,σ), which depends on the selection of the point a. There are many such examples [11]. From these examples we take an arbitrary one as our research object and this do not affect the proof and our results.

We construct a subsystem (X˜,σ) of (Σ3,σ) associated to the selected (X,σ) as follows.

For b=b0b1bnX, denote Xb={x=x0x1Σ3:nisuch thatxni=bi,andxn=2ifnni}, then Xb is

Conclusion

In summary, we showed some difference between the chaoticity of a compact system and the chaoticity of its induced hyperspace system. Devaney chaos in the hyperspace system need not implies Devaney chaos in its base system. Even if the hyperspace system is mixing, the former need not implies the latter. This kind of investigation should be useful in many real problems, such as in ecological modeling, demographic sciences associated to migration phenomena and numerical simulation, etc.

Acknowledgements

The first author is supported by the Youth Foundation of Institute of Mathematics, Jilin University. The second author was supported by a grant from Postdoctoral Science Research Program of Jiangsu Province (No. 0701049C) and Specialized Research Fund for Outstanding Young Teachers of East China University of Science and Technology. The first and third authors were supported by NSFC (No. 10771084).

References (12)

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