The persistence of ω-limit sets defined on compact spaces

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