Let $f: S^{1} \rightarrow S^{1}$ be an orientation-preserving invertible map of the circle onto itself, with a lift $F: \mathbb{R} \rightarrow \mathbb{R}$. Define the rotation numbers $\rho_{0}(F)$ and $\rho(f)$.

Suppose that $\rho_{0}(F)=p / q$, where $p$ and $q$ are coprime integers. Prove that the map $f$ has periodic points of least period $q$, and no periodic points with any least period not equal to $q$.

Now suppose that $\rho_{0}(F)$ is irrational. Explain the distinction between wandering and non-wandering points under $f$. Let $\Omega(x)$ be the set of limit points of the sequence $\left\{x, f(x), f^{2}(x), \ldots\right\}$. Prove

(a) that the set $\Omega(x)=\Omega$ is independent of $x$ and is the smallest closed, non-empty, $f$-invariant subset of $S^{1}$;

(b) that $\Omega$ is the set of non-wandering points of $S^{1}$;

(c) that $\Omega$ is either the whole of $S^{1}$ or a Cantor set in $S^{1}$.