On distributional chaos in non-autonomous discrete systems

TL;DR

This paper explores distributional chaos in non-autonomous discrete systems, establishing criteria for chaos based on properties like weak mixing and shadowing, with results linking different types of chaos in compact metric spaces.

Contribution

It provides new criteria for distributional chaos in non-autonomous systems, including equivalences and conditions involving topological and shadowing properties.

Findings

Li-Yorke{ extquoteright}chaos is equivalent to distributional chaos in compact spaces.

Three criteria for distributional chaos are established based on mixing, shadowing, and expansion.

A criterion for chaos induced by Xiong chaotic sets is developed.

Abstract

This paper studies distributional chaos in non-autonomous discrete systems generated by given sequences of maps in metric spaces. In the case that the metric space is compact, it is shown that a system is Li-Yorke{\delta}-chaotic if and only if it is distributionally{\delta}'-chaotic in a sequence; and three criteria of distributional {\delta}-chaos are established, which are caused by topologically weak mixing, asymptotic average shadowing property, and some expanding condition, respectively, where {\delta} and {\delta}' are positive constants. In a general case, a criterion of distributional chaos in a sequence induced by a Xiong chaotic set is established.