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Typical continuous function without cycles is stable. (English) Zbl 0582.54026
Let C be the metric space of all \(I\to I\) continuous functions with the uniform metric. If \(f\in C\) has a cycle of order n at x, the length of this cycle is \(d=\max \{| f^ r(x)-f^ s(x)|;\quad 1\leq r,s\leq n\}.\) Let \(\lambda\) (f) be the lowest upper bound of the lengths of all cycles of f. The function f is said to be stable if \(\lambda\) : \(C\to {\mathbb{R}}\) is continuous at f. Denoting by \(A=\{f\in C\); f is without cycles\(\}\) and by \(F=\{f\in A\); f is unstable\(\}\) the author proves that A is a second Baire category set in itself, while F is a first category set in A.
Reviewer: Gh.Toader
MSC:
54H20 Topological dynamics (MSC2010)
26A18 Iteration of real functions in one variable
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References:
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