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This question already has an answer here:

I need to show that if $f:S^{1} \rightarrow S^{1}$ is a preserving-order diffeomorphism and $f$ has irrational rotation number, then $f$ has at least one recurrent point. How can I prove that?

Definition: $x_0$ is a recurrent point under $f$ if, for any neighborhood $U$ of $x_0$, there exists $n>0$ such that $f^{n}(x_0) \in U $.

Edit: It is known that a preserving-order diffeomorphism has irrational rotation number if and only if it has no periodic points.

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marked as duplicate by Mark McClure, Mark Bennet, mrf, user147263, colormegone Jun 3 '15 at 21:50

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ I voted to close this since it's exactly the same as your more recent (and now answered) question here. $\endgroup$ – Mark McClure Jun 3 '15 at 18:40
  • $\begingroup$ Ok. I tried to delete but I don't know how to do that. Don't hate me, please. $\endgroup$ – mathgccunha Jun 3 '15 at 21:35
  • $\begingroup$ No biggie - my comment was just informative. :) $\endgroup$ – Mark McClure Jun 3 '15 at 21:39
  • $\begingroup$ @MarkMcClure I know, I was just kidding. $\endgroup$ – mathgccunha Jun 4 '15 at 0:19