Distributional chaotic generalized shifts

TL;DR

This paper characterizes when generalized shift maps on finite discrete spaces exhibit various forms of distributional chaos based on properties of the underlying map, providing a complete classification with counterexamples.

Contribution

It offers a full characterization of distributional chaos types for generalized shifts in terms of properties of the map ta, including non-quasi-periodic and one-to-one conditions.

Findings

Distributional chaos occurs if ta has a non-quasi-periodic point.

Dense distributional chaos occurs if ta has no periodic points.

Transitive distributional chaos occurs if ta is one-to-one without periodic points.

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.