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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.

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This questions seems borderline for this site, but let me give you hint : What happens when you look at the interval $[x,x'[$ where $x$ is any point and $x'$ is the closest point forward that lies in the orbit of $x$, and then at iterates of $x+\varepsilon$ for small positive $\varepsilon$? –  Benoît Kloeckner May 28 '13 at 18:08
    
Thank you for your hint. It is the first time I ask a question on this site, so can you explain what it seems borderline ? Thanks in advance? –  Maxence May 29 '13 at 11:15
    
@Maxence: You could consider accepting the answer if it was helpful to you. The question was borderline because it is about a fairly classical fact that you should have found in introductory books on the topic… –  Loïc Teyssier May 29 '13 at 16:35
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@BenoîtKloeckner I don't agree with the "borderline" label. Nobody is assumed to know everything on every subject, even on its own subject. I appreciate when specialists give their input even for some "naive" questions because sometime (depending on the qualities of the specialist) it can be enlightening. –  Patrick I-Z Dec 8 '13 at 10:45
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This exchange reminds me of a classic Andre Weil anecdote. A mathematician addresses Weil: "Can I ask you a stupid question?" - "You just did", Weil snaps back. –  Victor Protsak Dec 8 '13 at 21:04
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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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Thank you for your answer ! –  Maxence May 29 '13 at 11:13
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