I am sorry because it is probably a trivial question. I consider a homeomorphism of the circle that preserves orientation and that has a rational rotation number p/q (it is an irreducible fraction). I assume that ALL the orbits are periodic. Then we know that all the orbits have the same period q. I want to know if such a homeomorphism is conjugate to the rotation of angle p/q. Is that true ? If yes, can you tell me how to construct such a conjugacy ? I tried but I failed... If it is not true, can you give me a counterexample ? Thanks in advance.

## 1 Answer

The answer is "yes". Consider $\mathbb S^1$ as the quotient $\mathbb R/\mathbb Z$. Your homeomorphism $f$ lifts to a homeomorphism $\phi : \mathbb R \to \mathbb R$ such that $\phi(x+1)=\phi(x)+1$. Form the map $h:=\frac{1}{q} \sum _{n=1} ^q (\phi^{\circ n}-pn)$, where $\phi ^{\circ n}$ is the composition $n$ times of $\phi$ with itself. By construction $h\circ \phi = h+\frac{p}{q} $ and $h(x+1)=1+h(x)$, so that $h$ factors as a homeomorphism of the circle conjugating $f$ to the rotation. By the way this approach wors in $\mathbb R^n$ too.

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