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On a theorem by Florek and Slater on recurrence properties of circle maps. (English) Zbl 0713.11044

Let \(\beta\) be real and \(0<a\leq 1\). Then the theorem by Florek and Slater says that the gaps between the successive values of n for which \(n\beta\) mod 1\(<a\) can have at most three lengths [cf. N. B. Slater, Proc. Camb. Philos. Soc. 63, 1115-1123 (1967; Zbl 0178.047)]. The authors show that this result holds also for the gaps of \(-a<n\beta\) mod 1\(<a\). An application is given to the recurrence behaviour of the unit circle under rotations by the angle \(\beta\).
Reviewer’s remark: Using only one sheet it can easily be shown that the theorem by Florek and Slater holds for the gaps of \(a\leq n\beta mod 1<b\) for any \(a<b\).
Reviewer: P.Schatte

MSC:

11K06 General theory of distribution modulo \(1\)
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics

Citations:

Zbl 0178.047
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References:

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