Iterative Stability in the Class of Continuous Functions

Abstract

Let $\mathcal{K}$ be the class of compact subsets of $I=[0,1]$, and $\mathcal{K}^{*}$ consist of the nonempty closed subsets of $\mathcal{K}$. We study the maps $\Lambda :C(I,I)\rightarrow \mathcal{K}$ and $\Omega :C(I,I)\rightarrow \mathcal{K}^{*}$ defined so that $\Lambda (f)$ is the set of $\omega $-limit points of $f$, and $\Omega \bigl( f\bigr) $ is the collection of $\omega $-limit sets of $f$. We find that, in general, neither map is continuous. We do get more positive results, however, if we restrict ourselves to certain types of $\omega $-limit sets and better behaved classes of functions. We find that when $f$ has only a finite number of $\omega $-limit sets, each demonstrating a certain type of stability, the function $\Omega $ is continuous at $f$. The map $\Omega \mid \mathcal{E}$ is also studied, where $\mathcal{E}$ is composed of those continuous functions with zero topological entropy, and a significant degree of stability of $\Omega \mid \mathcal{E}$ is established.