Let $F: I \rightarrow I$ be a continuous one-dimensional map of an interval $I \subset \mathbb{R}$. Explain what is meant by saying (a) that $F$ has a horseshoe, (b) that $F$ is chaotic (Glendinning's definition).

Consider the tent map defined on the interval $[0,1]$ by

$$F(x)= \begin{cases}\mu x & 0 \leqslant x<\frac{1}{2} \\ \mu(1-x) & \frac{1}{2} \leqslant x \leqslant 1\end{cases}$$

with $1<\mu \leqslant 2$.

Find the non-zero fixed point $x_{0}$ and the points $x_{-1}<\frac{1}{2}<x_{-2}$ that satisfy

$$F^{2}\left(x_{-2}\right)=F\left(x_{-1}\right)=x_{0} .$$

Sketch a graph of $F$ and $F^{2}$ showing the points corresponding to $x_{-2}, x_{-1}$ and $x_{0}$. Hence show that $F^{2}$ has a horseshoe if $\mu \geqslant 2^{1 / 2}$.

Explain briefly why $F^{4}$ has a horseshoe when $2^{1 / 4} \leqslant \mu<2^{1 / 2}$ and why there are periodic points arbitrarily close to $x_{0}$ for $\mu \geqslant 2^{1 / 2}$, but no such points for $2^{1 / 4} \leqslant \mu<2^{1 / 2}$.