The three versions of distributional chaos

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Abstract

The notion of distributional chaos was introduced by Schweizer and Smı́tal [Trans. Amer. Math. Soc. 344 (1994) 737] for continuous maps of the interval. However, it turns out that, for continuous maps of a compact metric space three mutually nonequivalent versions of distributional chaos, DC1–DC3, can be considered. In this paper we consider the weakest one, DC3. We show that DC3 does not imply chaos in the sense of Li and Yorke. We also show that DC3 is not invariant with respect to topological conjugacy. In other words, there are lower and upper distribution functions Φxy and Φxy* generated by a continuous map f of a compact metric space (M, ρ) such that Φxy*(t)>Φxy(t) for all t in an interval. However, f on the same space M, but with a metric ρ′ generating the same topology as ρ is no more DC3.

Recall that, contrary to this, either DC1 or DC2 is topological conjugacy invariant and implies Li and Yorke chaos (cf. [Chaos, Solitons & Fractals 21 (2004) 1125]).

Section snippets

Introduction and preliminaries

Let f be a map from a compact metric space (M, ρ) into itself. For any pair (x, y) of points in M and any positive integer n, define a distribution function Φxy(n):R[0,1] byΦxy(n)(t)=1n#{0in-1;ρ(fi(x),fi(y))<t}.Then Φxy(n)(t) is a non-decreasing function, Φxy(n)(t)=0 for t  0, and Φxy(n)(t)=1 for t greater than the diameter of M. PutΦxy(t)=liminfnΦxy(n),andΦxy*(t)=limsupnΦxy(n)(t).Then Φxy is called the lower distribution function, and Φxy* the upper distribution function of x and y.

Proof of the main results

Theorem 1

There is a continuous map F of a compact metric space M which is DC3 but not chaotic in the sense of Li and Yorke.

Proof

Let X be the set {0, 1} of sequences of two symbols, equipped with the metric ρX of pointwise convergence. Let Y be the subset of the complex plane consisting of the unit circle, and a singleton {2}; i.e., z  Y if and only if either ∣z = 1 or z = 2. Finally, let M = X × Y, and let ρ be the max metric on M given by ρ((x1, y1), (x2, y2)) = max{ρX(x1, x2), ∣y1  y2∣}. For any (x, y)  M we let F(x, y

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The research was supported, in part, by projects 201/03/1153 from the Grant Agency of Czech Republic, MSM192400002 from the Czech Ministry of Education, and PI-8/00807/FS/01 from fundación Séneca (Comunidad Autonóma de Murcia).

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