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