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1998/1999 Iterative Stability in the Class of Continuous Functions
T. H. Steele
Real Anal. Exchange 24(2): 765-780 (1998/1999).


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.


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T. H. Steele. "Iterative Stability in the Class of Continuous Functions." Real Anal. Exchange 24 (2) 765 - 780, 1998/1999.


Published: 1998/1999
First available in Project Euclid: 28 September 2010

zbMATH: 0967.26005
MathSciNet: MR1704748

Primary: 26A18

Keywords: iterative stability

Rights: Copyright © 1999 Michigan State University Press

Vol.24 • No. 2 • 1998/1999
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