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# Prove that the rotation number exists and is well-defined

This is commonly taught in the first semester of a dynamical systems course. Say we have a continuous, increasing function $$F : \mathbb{R}\to \mathbb{R}$$ with the property that $$F(x+1) = F(x)+1$$ for all $$x\in \mathbb{R}$$. Prove that $\omega(F) = \lim_{n\to \infty} \frac{F^n(x)}{n}$ exists and is independent of $$x$$. Here, $$F^n(x) = (F\circ F\circ\cdots \circ F)(x)$$, with $$n$$ copies of $$F$$ on the right-hand side.

Note by Michael Lee
4 months ago