Taken from these notes [1] on Galois Theory, I would like to show that iterating the map
$$p: x \mapsto x^2 - x - 2 $$
has a cycle of order 3 when you start with the root of $x^3 - 3x - 1 = 0$.
It would be interesting to find more cycles. Here it is interesting the degree 8 polynomial $(p\circ p \circ p)(x) =x$ has a cubic factor.
Wolfram Alpha [2] says it factors into two cubics and a quadratic:
$$(p\circ p \circ p)(x) - x = (x^3 - 3x - 1 )(x^2 - 2x - 2)(x^3 -2x^2 -3x + 5) $$
In those same notes, we learn the polynomial $x^3 - a(x-1) = 0$ also has Galois group $A_3 = C_3$ when $a = k^2 + k + 7$. What is the quadratic substitution that generates it?