Let $\mathcal{S}$ be a metric space, $F$ a map of $\mathcal{S}$ to itself and $P$ a point of $\mathcal{S}$. Define an attractor for $F$ and an omega point of the orbit of $P$ under $F$.

Let $f$ be the map of $\mathbb{R}$ to itself given by

$$f(x)=x+\frac{1}{2}+c \sin ^{2} 2 \pi x$$

where $c>0$ is so small that $f^{\prime}(x)>0$ for all $x$, and let $F$ be the map of $\mathbb{R} / \mathbb{Z}$ to itself induced by $f$. What points if any are

(a) attractors for $F^{2}$,

(b) omega points of the orbit of some point $P$ under $F$ ?

Is the cycle $\left\{0, \frac{1}{2}\right\}$ an attractor?

In the notation of the first two sentences, let $\mathcal{C}$ be a cycle of order $M$ and assume that $F$ is continuous. Prove that $\mathcal{C}$ is an attractor for $F$ if and only if each point of $\mathcal{C}$ is an attractor for $F^{M}$.