# Existence of a recurrent point [duplicate]

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.