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

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

CONTINUITY OF THE MAPS f →⋃x∈Iω(x, f) AND f → {ω(x, f) : x ∈ I}

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∈I ω(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. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved. 1