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 $\varphi:\Gamma\to\Gamma$. We prove that the generalized shift $\sigma_\varphi:X^\Gamma\to X^\Gamma$ with $\sigma_\varphi((x_\alpha)_{\alpha\in\Gamma})=(x_{\varphi(\alpha)})_{\alpha\in\Gamma}$ (for $(x_\alpha)_{\alpha\in\Gamma}\in X^\Gamma$) is: $\bullet$ distributional chaotic (uniform, type 1, type 2) if and only if $\varphi:\Gamma\to\Gamma$ has at least a non-quasi-periodic point, $\bullet$ dense distributional chaotic if and only if $\varphi:\Gamma\to\Gamma$ does not have any periodic point, $\bullet$ transitive distributional chaotic if and only if $\varphi:\Gamma\to\Gamma$ is one--to--one without any periodic point. We complete the text by counterexamples.