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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$\begingroup$ 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 $\varepsilon$? $\endgroup$– Benoît KloecknerCommented May 28, 2013 at 18:08
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$\begingroup$ 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? $\endgroup$– MaxenceCommented May 29, 2013 at 11:15
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1$\begingroup$ @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. $\endgroup$– Patrick I-ZCommented Dec 8, 2013 at 10:45
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1$\begingroup$ @PatrickI-Z: I find this question borderline not because everybody should know the answer, but because it seemed to me that any mathematician could find the answer, or at least explain what he or she tried in the question. And borderline does not mean clearly out of scope. $\endgroup$– Benoît KloecknerCommented Dec 8, 2013 at 20:30
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3$\begingroup$ 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. $\endgroup$– Victor ProtsakCommented Dec 8, 2013 at 21:04
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1 Answer
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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 works in $\mathbb R^n$ too.
answered May 28, 2013 at 18:10
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$\begingroup$ Loïc Teyssier, could you tell me how you created this function? I dont understand $\endgroup$ Commented Oct 21 at 12:06
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$\begingroup$ I didn't create it, it is just a standard averaging method in dynamical systems. $\endgroup$ Commented Oct 21 at 14:47