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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.