Exercise 7.13 Consider the logistic map
$x_{n+1} = 4\mu x_n (1 - x_n)$,
where $x_n \in [0, 1]$, and $\mu = 1$. Show that the solutions may be written as $x_n = \sin^2 \theta_n$
where $\theta^n \in \mathbb{T}$, and
$\theta_{n+1} = 2\theta_n$.
What can you say about the orbits of the logistic map, the existence of an invariant
measure, and the validity of an ergodic theorem?