The persistence of ω-limit sets defined on compact spaces

Let K be the class of closed subsets of a compact metric space X, and K ⋆ consist of the nonempty closed subsets of K . We study the maps f ↦ L ( f ) and f ↦ L ⋆ ( f ) defined so that L ( f ) is the collection of ω-limit sets of f, and L ⋆ ( f ) = { L ⊆ X : L is closed , f ( L ) = L , and F ∩ f ( L ∖ F ) ¯ ≠ ∅ whenever F is a nonempty and proper closed subset of L } . We show that L ⋆ ( f ) is always closed in K , hence L ⋆ ( f ) ∈ K ⋆ , and that the map L ⋆ : C ( X , X ) → K ⋆ is upper semicontinuous. Using the notion of a periodic orbit stable under perturbation, we give a sufficient condition on f for L ⋆ to be continuous there, and establish a residual subset of C ( M , M ) , when M is an n-manifold with n ⩾ 1 , where L ⋆ is continuous. These results on L ⋆ are fundamental to our study of the map L . We characterize those f in C ( I , I ) at which L is continuous, and show that L is continuous on a residual subset of C ( I , I ) . Similarly, the map f ↦ L ( f ) ¯ is continuous on a residual subset of C ( M , M ) , and we characterize those functions in C ( M , M ) at which f ↦ L ( f ) ¯ is upper semicontinuous.