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

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.