Let $I, J$ be closed bounded intervals in $\mathbb{R}$, and let $F: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous map.

Explain what is meant by the statement that ' $I F$-covers $J$ ' (written $I \rightarrow J)$. For a collection of intervals $I_{0}, \ldots, I_{k}$ define the associated directed graph $\Gamma$ and transition matrix $A$. Derive an expression for the number of (not necessarily least) period- $n$ points of $F$ in terms of $A$.

Let $F$ have a 5 -cycle

$$x_{0}<x_{1}<x_{2}<x_{3}<x_{4}$$

such that $x_{i+1}=F\left(x_{i}\right)$ for $i=0, \ldots, 4$ where indices are taken modulo 5 . Write down the directed graph $\Gamma$ and transition matrix $A$ for the $F$-covering relations between the intervals $\left[x_{i}, x_{i+1}\right]$. Compute the number of $n$-cycles which are guaranteed to exist for $F$, for each integer $1 \leqslant n \leqslant 4$, and the intervals the points move between.

Explain carefully whether or not $F$ is guaranteed to have a horseshoe. Must $F$ be chaotic? Could $F$ be a unimodal map? Justify your answers.

Similarly, a continuous map $G: \mathbb{R} \rightarrow \mathbb{R}$ has a 5 -cycle

$$x_{3}<x_{1}<x_{0}<x_{2}<x_{4}$$

For what integer values of $n, 1 \leqslant n \leqslant 4$, is $G$ guaranteed to have an $n$-cycle?

Is $G$ guaranteed to have a horseshoe? Must $G$ be chaotic? Justify your answers.