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Let $f,g:S^1 \to S^1$ be orientation-preserving homeomorphisms. Consider the lift $F,G:\mathbb R \to \mathbb R$. Let $\rho(G)$ and $\rho(F)$ be a rotation numbers. What we can say about rotation number of their composition? i.e about $\rho(F\circ G)$. I know that if $F$ and $G$ are commute then $\rho(F\circ G)=\rho(F)+\rho(G)$. But what if not? For example if $\rho(G) = \rho(F)=0$ then what can we say about $\rho(F\circ G)$? I suppose that this will be a rational number from $0$ to $1$, but how to show that or maybe it's not true, I also think that I need to use that if we have 2 homeomorphism of circle then they are conjugate, i.e there exist $h$ such that $f\circ h=h\circ g$

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