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

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.