Continuity of the maps f↦∪x∈Iω(x,f) and f↦{ω(x,f):x∈I}

Abstract

We study the behavior of two maps in an effort to better understand the stability of ω-limit sets ω(x,f) as we perturb either x or f, or both. The first map is the set-valued function Λ taking f in C(I,I) to its collection of ω-limit points Λ(f)=∪x∈Iω(x,f), and the second is the map Ω taking f in C(I,I) to its collection of ω-limit sets Ω(f)={ω(x,f):x∈I}. We characterize those functions f in C(I,I) at which each of our maps Λ and Ω is continuous, and then go on to show that both Λ and Ω are continuous on a residual subset of C(I,I). We then investigate the relationship between the continuity of Λ and Ω at some function f in C(I,I) with the chaotic nature of that function.