It is well known that ω-limit sets are internally chain transitive and have weak incompressibility; the converse is not generally true, in either case. However, it has been shown that a set is weakly incompressible if and only if it is an abstract ω-limit set, and separately that in shifts of finite type, a set is internally chain transitive if and only if it is a (regular) ω-limit set. In this paper we generalise these and other results, proving that the characterization for shifts of finite type holds in a variety of topologically hyperbolic systems (defined in terms of expansive and shadowing properties), and also show that the notions of internal chain transitivity and weak incompressibility coincide in compact metric spaces.2000 Mathematics Subject Classification. 37B25, 37B45, 37E05, 54F15, 54H20.
Abstract. We address various notions of shadowing and expansivity for continuous maps restricted to a proper subset of their domain. We prove new equivalences of shadowing and expansive properties, we demonstrate under what conditions certain expanding maps have shadowing, and generalize some known results in this area. We also investigate the impact of our theory on maps of the interval.
A set Λ is internally chain transitive if for any x,y∈Λ and ϵ>0 there is an ϵ-pseudo-orbit in Λ between x and y. In this paper we characterize all ω-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 point z∈X with ω(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 the ω-limit set of some point in the full shift space over ℬ. We use similar techniques to prove that, for a tent map f, a closed, strongly f-invariant, internally chain transitive subset of the interval is the ω-limit set of a point provided it does not contain the image of the critical point. We give an example of a sofic shift space Z𝒢 (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 the ω-limit set of any point in Z𝒢.
Abstract. We address various notions of shadowing and expansivity for continuous maps restricted to a proper subset of their domain. We prove new equivalences of shadowing and expansive properties, we demonstrate under what conditions certain expanding maps have shadowing, and generalize some known results in this area. We also investigate the impact of our theory on maps of the interval.
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