## 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.Bronštein

### MSC:

 37B99 Topological dynamics

rotation numbers
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