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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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