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 $Γ$, and $\varphi:Γ\toΓ$ we prove the generalized shift dynamical system $(X^Γ,σ_\varphi)$ is densely chaotic if and only if $\varphi:Γ\toΓ$ does not have any (quasi-)periodic point. Hence the class of all densely chaotic generalized shifts on $X^Γ$ is intermediate between the class of all Devaney chaotic generalized shifts on $X^Γ$ and the class of all Li-Yorke chaotic generalized shifts on $X^Γ$. In addition, these inclusions are proper for infinite countable $Γ$. Moreover we prove $(X^Γ,σ_\varphi)$ 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 $\varphi:Γ\toΓ$ has at least one non-quasi-periodic point.

Theorems & Definitions (25)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Remark 3.3
• Lemma 3.4
• proof
• Lemma 3.5
• proof
• Theorem 3.6: Densely chaotic generalized shifts