Distributional chaos revisited

Abstract

Abstract. In their famous paper, Schweizer and Smítal introduced the definition of a distributionally chaotic pair and proved that the existence of such a pair implies positive topological entropy for continuous mappings of a compact interval. Further, their approach was extended to the general compact metric space case.In this article we provide an example which shows that the definition of distributional chaos (and as a result Li-Yorke chaos) may be fulfilled by a dynamical system with (intuitively) regular dynamics embedded in R 3 . Next, we state strengthened versions of distributional chaos which, as we show, are present in systems commonly considered to have complex dynamics.We also prove that any interval map with positive topological entropy contains two invariant subsets X, Y ⊂ I such that f | X has positive topological entropy and f | Y displays distributional chaos of type 1, but not conversely.

Distributional chaos for backward shifts

“…We will consider in this paper only the definition of uniform distributional chaos, which is one of the strongest possibilities [43]. This property can be defined in the following way:…”

Distributional chaos for linear operators

“…For each pair x, y ∈ X and each n ∈ N, the distributional function F The following notions were introduced in [23] and [21], respectively. They were considered for linear operators on Banach or Fréchet spaces in [6,12,13,14,16,17,20,24].…”

Topological entropy for local processes

“…We can also assume that entropy of f is positive and X is compact, since such example can be easily constructed [19]. It is also known, that such a map has unique fixed point, let…”