A characterization of ω-limit sets in shift spaces

A set A is internally chain transitive if or any x, y is an element of A and epsilon > 0 there is an epsilon-pseudo-orbit in Lambda between x and y. In this paper we characterize all omega-limit sets in shifts of finite type by showing that, if Lambda is a closed, strongly shift-invariant subset of a shift of finite type, X, then there is a point z is an element of X with omega(z) = Lambda if and only if Lambda 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 omega-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 omega-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 Z(G) (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 omega-limit set of any point in Z(G).