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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
1 month, 4 weeks ago

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