A characterization ofω-limit sets in shift spaces

Abstract

AbstractA set Λ isinternally chain transitiveif for anyx,y∈Λ and ϵ>0 there is an ϵ-pseudo-orbit in Λ betweenxandy. In this paper we characterize allω-limit sets in shifts of finite type by showing that, if Λ is a closed, strongly shift-invariant subset of a shift of finite type,X, then there is a pointz∈Xwithω(z)=Λ if and only if Λ is internally chain transitive. It follows immediately that any closed, strongly shift-invariant, internally chain transitive subset of a shift space over some alphabet ℬ is theω-limit set of some point in the full shift space over ℬ. We use similar techniques to prove that, for a tent mapf, a closed, stronglyf-invariant, internally chain transitive subset of the interval is theω-limit set of a point provided it does not contain the image of the critical point. We give an example of a sofic shift spaceZ𝒢(a factor of a shift space of finite type) that is not of finite type that has an internally chain transitive subset that is not theω-limit set of any point inZ𝒢.