International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 82623, 15 pages

Continuity of the maps fxIω(x,f) and f{ω(x,f):xI}

T. H. Steele

Department of Mathematics, Weber State University, Ogden 84408-1702, UT, USA

Received 29 September 2004; Revised 11 December 2005; Accepted 12 February 2006

Copyright © 2006 T. H. Steele. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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)=xIω(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):xI}. 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.