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 $\phi :\Gamma \to \Gamma$ we prove the generalized shift dynamical system $(X^{\Gamma },{\sigma }_{\phi })$ is densely chaotic if and only if $\phi :\Gamma \to \Gamma$ 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 $(X^{\Gamma },{\sigma }_{\phi })$ 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 $\phi :\Gamma \to \Gamma$ has at least one non-quasi-periodic point.

14 pages - regarding some errors/mistakes please don't use previous versions