Rotation number of composition 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 $\rho(F\circ G)$? 
 A: Write $S^1=\mathbf{R}/\mathbf{Z}$. Denote by $\mathcal{G}$ of increasing self-homeomorphisms of $\mathbf{R}$ commuting with the translation $z\mapsto z+1$, that is, whose graph is invariant by the translation $(x,y)\mapsto$ $(x+1,y+1)$. So the rotation number $\rho(F)$ can be defined as $$\rho(F)=\lim_{n\to+\infty}\frac{F^n(x)-x}{n},$$ which does not depend on $x$. Your question amounts to ask which values of $\rho(G\circ F)$ are possible when $\rho(F)=\rho(G)=0$:

The answer is: $[-1,1]$.

Indeed, since $\rho(F)=\rho(G)=0$, they have fixed points $x_F,x_G$, we can suppose $x_F\in [x_G,x_G+1\mathclose[$. Fix $x\le x_F$. Then $F(x)\le x_F\le x_G+1$, and in turn $G\circ F(x)\le x_G+1\le x_F+1$. By induction, $(G\circ F)^n(x)\le x_F+n$. This shows that $\rho(G\circ F)\le 1$. The inequality $\ge -1$ follows by a similar argument, or applying this to the inverse.
Conversely, all values in $[-1,1]$ are attained:


*

*how to obtain $\pm 1$: Consider $F$ fixing 0 and mapping $1/4$ to $3/4$. Conjugating by the translation $+1/2$, we define $G$, fixing $1/2$, and mapping $3/4$ to $5/4$. Hence $G\circ F$ maps $1/4$ to $5/4=1+1+4$, which implies that $\rho(G\circ F)=1$. Passing to the inverse, we also obtain $-1$. 

*how to obtain $t$ with $-1<t<1$: Let $R_t$ be the translation by $t$. Choose $F$ with the prescription that $F(0)=0$ and that there exists $x$ with $F(x)=x+t$ (this is possible because $|t|<1$). Define $G=R_t\circ F^{-1}$. Then $G(x+t)=R_t\circ F^{-1}(x+t)=R_t(x)=x+t$, so $\rho(F)=\rho(G)=0$, while $\rho(G\circ F)=\rho(R_t)=t$. 
A: @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 $\rho(G)$.
