A set Λ is internally chain transitive if for any x,yΛ and >0 there is an -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 zX 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 is the ω-limit set of some point in the full shift space over . 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.
|Translated title of the contribution||A characterization of omega-limit sets in shift spaces|
|Pages (from-to)||21 - 31|
|Number of pages||11|
|Journal||Ergodic Theory and Dynamical Systems|
|Early online date||26 Feb 2009|
|Publication status||Published - 1 Feb 2010|