16. Rational Rotation Numbers

We begin by showing that if $f \colon S^1 \to S^1$ is an orientation-preserving reversible dynamical system then $\mathsf{per}(f) \ne \emptyset$ if and only if $\operatorname{rot}(f) \in\mathbb{Q}$.  

We then focus on the (ultimately less interesting case) where the rotation number is rational. In this case the dynamics turn out to be essentially trivial: any orbit is asymptotic to a periodic orbit and any two periodic orbits have the same period.

Next week we will turn to the more exciting case of irrational rotation numbers.