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Iterative stability in the class of continuous functions. (English) Zbl 0967.26005

Let \(\mathcal K\) be the class of non-empty compact subsets of the interval \(I\), and let \(\mathcal K^*\) consist of the non-empty subsets of \(\mathcal K\) closed in the Hausdorff metric. The author considers the maps \(\Lambda :C(I,I)\rightarrow \mathcal K\) and \(\Omega :C(I,I)\rightarrow \mathcal K^*\) defined such that \(\Lambda (f)\) is the set of \(\omega \)-limit points of \(f\), and \(\Omega (f)\) is the collection of \(\omega \)-limit sets of \(f\). He looks for subclasses of \(C(I,I)\) on which the mappings \(\Lambda \) and \(\Omega \) are continuous. Sample result: Assume \(f_n\rightarrow f\) uniformly in \(C(I,I)\), \(\omega _n\rightarrow \omega \) in \(\mathcal K\), and \(\omega _n\in \Omega (f_n)\) for each \(n\). (i) If \(\omega \) is a finite set, then \(\omega \in \Omega (f)\). (ii) If each \(f_n\) has zero topological entropy and \(\omega \) is infinite, then the set of non-isolated points of \(\omega \) belongs to \(\Omega (f)\). The paper contains interesting examples showing that the mappings \(\Lambda \) and \(\Omega \) are strongly discontinuous.

MSC:

26A18 Iteration of real functions in one variable
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