TITLE: Rotation number of composition
QUESTION [5 upvotes]: 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 $\rho(F\circ G)$?

REPLY [3 votes]: @user3421341,Theorem 3.9 in the paper of Calegari and Walker gives a complete answer to your question, for arbitrary values of $\rho(F)$ and $\rho(G)$.