On a class between Devaney chaotic and Li-Yorke chaotic generalized shift dynamical systems

TL;DR

This paper characterizes when generalized shift dynamical systems are densely chaotic, placing them between Devaney and Li-Yorke chaos, and links various sensitivity properties to the existence of non-quasi-periodic points.

Contribution

It establishes a precise criterion for dense chaos in generalized shifts and clarifies the relationship between different chaos classes based on the properties of the function ta.

Findings

Densely chaotic shifts occur iff ta has no (quasi-)periodic points.

The class of densely chaotic shifts is strictly between Devaney and Li-Yorke chaotic classes.

Various sensitivity properties are equivalent to the existence of at least one non-quasi-periodic point.

Abstract

In the following text, for finite discrete $X$ with at least two elements, nonempty countable $\Gamma$, and $\varphi:\Gamma\to\Gamma$ we prove the generalized shift dynamical system $(X^\Gamma,\sigma_\varphi)$ is densely chaotic if and only if $\varphi:\Gamma\to\Gamma$ does not have any (quasi-)periodic point. Hence the class of all densely chaotic generalized shifts on $X^\Gamma$ is intermediate between the class of all Devaney chaotic generalized shifts on $X^\Gamma$ and the class of all Li-Yorke chaotic generalized shifts on $X^\Gamma$. In addition, these inclusions are proper for infinite countable $\Gamma$. Moreover we prove $(X^\Gamma,\sigma_\varphi)$ is Li-Yorke sensitive (resp. sensitive, strongly sensitive, asymptotic sensitive, syndetically sensitive, cofinitely sensitive, multi-sensitive, ergodically sensitive, spatiotemporally chaotic, Li-Yorke chaotic) if and only if $\varphi:\Gamma\to\Gamma$ has at least one non-quasi-periodic point.