Let $F, G : \mathbb{R} \rightarrow \mathbb{R}$ be a lift of $f$ and $G$ of $g$. That is, $ \pi \circ F = f \circ \pi$ with $\pi(x) = e^{2\pi i}$. We define $$\rho_{0}(F) = \lim_{n\rightarrow \infty}\frac{F^{n}}{n}.$$ One property: for $F,F'$ two different lifts of $f$, $$|\rho_{0}(F) - \rho_{0}(F)| \in \mathbb{Z}.$$ And another: $G^{-1}\circ F \circ G$ is a lift for $g^{-1}\circ f \circ g$. The rotation number of $f$, $\rho(f)$, is the fraction part of $\rho_{0}(F)$ for any lift $F$ of $f$. That is, $\rho(f)$ is the unique number in $[0,1)$ such that $\rho_{0}(F) - \rho(f)$ is an integer.

Can you help me prove the title-statement?

I got the hint that $$\lim_{n\rightarrow\infty} \frac{(G^{-1} \circ F \circ G)^{n}(x)}{n} = \lim_{n\rightarrow\infty}\frac{G^{-1}\circ F^{n} \circ G(x)}{n}$$ and also that when this last expression equals $\rho_{0}(F)$ we have that the requested result follows. But how to finish this?


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