(a) Let $F: I \rightarrow I$ be a continuous map defined on an interval $I \subset \mathbb{R}$. Define what it means (i) for $F$ to have a horseshoe and (ii) for $F$ to be chaotic. [Glendinning's definition should be used throughout this question.]

(b) Consider the map defined on the interval $[-1,1]$ by

$$F(x)=1-\mu|x|$$

with $0<\mu \leqslant 2$.

(i) Sketch $F(x)$ and $F^{2}(x)$ for a case when $0<\mu<1$ and a case when $1<\mu<2$.

(ii) Describe fully the long term dynamics for $0<\mu<1$. What happens for $\mu=1$ ?

(iii) When does $F$ have a horseshoe? When does $F^{2}$ have a horseshoe?

(iv) For what values of $\mu$ is the map $F$ chaotic?