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.