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

Prove that if $F$ has a 3 -cycle $x_{1}<x_{2}<x_{3}$ then $F$ is chaotic. [You may assume the intermediate value theorem and any corollaries of it.]

State Sharkovsky's theorem.

Use the above results to deduce that if $F$ has an $N$-cycle, where $N$ is any integer that is not a power of 2 , then $F$ is chaotic.

Explain briefly why if $F$ is chaotic then $F$ has $N$-cycles for many values of $N$ that are not powers of 2. [You may assume that a map with a horseshoe acts on some set $\Lambda$ like the Bernoulli shift map acts on $[0,1)$.]

The logistic map is not chaotic when $\mu<\mu_{\infty} \approx 3.57$ and it has 3 -cycles when $\mu>1+\sqrt{8} \approx 3.84$. What can be deduced from these statements about the values of $\mu$ for which the logistic map has a 10-cycle?