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Dynamical stability of the typical continuous function. (English) Zbl 1150.26002
Studying the iterative stability of \(f\in C(I,I)\) we can consider the union of all \(\omega \) limit sets \(\Lambda (f)=\bigcup \omega (x,f)\) as well as the system of all \(\omega \) limit sets \(\Omega (f)=\{\omega (x,f)\:x\in I \}\). If we take corresponding maps \(\Lambda \: C(I,I)\to (K,H)\) and \(\Omega \: C(I,I) \to (K^*,H^*)\) where \((K,H)\) is the class of nonempty closed sets \(K\) in \(I\) endowed with Hausdorff metric \(H\) and \((K^*,H^*)\) is the system of nonempty closed subsets of \(K\) there are several results characterizing points of continuity of \(\Lambda \) and \(\Omega \). Present paper shows that both \(\Lambda \) and \(\Omega \) are continuous on a residual subset of \(C(I,I)\). Further it is shown that the chaotic nature of \(f\) does not depend in general on the continuity of these maps.

MSC:
26A18 Iteration of real functions in one variable
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References:
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