Mixing, Li-Yorke chaos, distributional chaos and Kato's chaos to multiple mappings

TL;DR

This paper explores various types of chaos in multiple continuous mappings on compact metric spaces, establishing implications among them and providing constructions to illustrate the generation of chaos from minimal conditions.

Contribution

It introduces new definitions of transitivity, weakly mixing, and mixing for multiple mappings and links these to different chaos concepts, also proving equivalence and constructing examples.

Findings

Mixing implies distributional, Li-Yorke, and Kato's chaos.

Kato's chaos is preserved under iteration.

Distributional chaos can be generated by only two strongly non-wandering points.

Abstract

Let $(X,d)$ be a compact metric space and $F=\{f_1,f_2,...,f_m\}$ be an $m$-tuple of continuous maps from $X$ to itself. In this paper, we introduce the definitions of transitivity, weakly mixing and mixing of multiple mappings $(X,F)$ from a set-valued perspective, which is the semigroup generated by $F$ based on iterated function system. Firstly, we prove that for multiple mappings, mixing implies distributional chaos in a sequence, Li-Yorke chaos and Kato's chaos. Besides, we demonstrate that $F$ is Kato's chaos if and only if $F^k$ is Kato's chaos for any $k \in \mathbb{N}$. Finally, we construct a symbolic dynamical system to show that distributional chaos may be generated by only two strongly non-wandering points.