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

Let $(X,d)$ be a compact metric space and F=₁,f₂,...,fₘ\ 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 Fᵏ is Kato's chaos for any k ∈ N. Finally, we construct a symbolic dynamical system to show that distributional chaos may be generated by only two strongly non-wandering points.