Distributional chaotic generalized shifts

Abstract

Suppose $X$ is a finite discrete space with at least two elements, $\Gamma$ is a nonempty countable set, and consider self--map $\phi :\Gamma \to \Gamma$. We prove that the generalized shift ${\sigma }_{\phi }:X^{\Gamma }\to X^{\Gamma }$ with ${\sigma }_{\phi }((x_{\alpha }{)}_{\alpha \in \Gamma })=(x_{\phi (\alpha )}{)}_{\alpha \in \Gamma }$ (for $(x_{\alpha }{)}_{\alpha \in \Gamma }\in X^{\Gamma }$) is: $\bullet$ distributional chaotic (uniform, type 1, type 2) if and only if $\phi :\Gamma \to \Gamma$ has at least a non-quasi-periodic point, $\bullet$ dense distributional chaotic if and only if $\phi :\Gamma \to \Gamma$ does not have any periodic point, $\bullet$ transitive distributional chaotic if and only if $\phi :\Gamma \to \Gamma$ is one--to--one without any periodic point. We complete the text by counterexamples.

arXiv comment13 pages