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Rotation numbers for monotone functions on the circle. (English) Zbl 0623.58008

Several results on rotation numbers for noncontinuous functions are obtained. Let \({\mathcal M}=\{F: {\mathbb{R}}\to {\mathbb{R}}|\) F is nondecreasing and \(F(t+1)\equiv F(t)+1\}\), \({\mathcal L}=\{F\in {\mathcal M}|\) F is strictly increasing\(\}\). Theorem 1. If \(F\in {\mathcal M}\) then \(\rho\) (F)\(\equiv \lim_{n\to \infty}F^ n(x)/n\) exists for all \(x\in {\mathbb{R}}\) and is independent of x.
Two functions F,G\(\in {\mathcal M}\) are called equivalent iff they are equal except at a countable set of points (of discontinuity). Theorem 3. Let \(F\in {\mathcal L}\), and p,q\(\in {\mathbb{Z}}\) with \(q>0\). Then \(\rho (F)=p/q\) iff there is a function \(K\in {\mathcal L}\) which is equivalent to F and is such that \(K^ q(x_ 0)=x_ 0+p\) for some \(x_ 0\in {\mathbb{R}}\).
Reviewer: I.U.Bron┼ítein

MSC:

37B99 Topological dynamics
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