TITLE: A problem about Cantor set and found when learning dynamical systems.
QUESTION [7 upvotes]: Consider the family of functions F(x)=$x^3 -\alpha$x,   for $\alpha \gt 0$
Prove that if $\alpha$ is sufficiently large, then the set of points |$F^n(x)$| which do not tend to infinity is a Cantor set.
Note:$F^n(x)$ means the iteration of the function; I have proved that if |$x$| is sufficiently large, then |$F^n(x)$|$\rightarrow\infty$
Your answeres will be greatly appreciated!
( This problem is from < An introduction to chaotic dynamical systems > Robert L. Devaney )

REPLY [2 votes]: You are trying to show that the filled Julia set is a Cantor set. This is equivalent to showing that the Julia set is a Cantor set. Now, the Julia set is a Cantor set if the critical points do not lie in the filled Julia set. So you can just check that the critical point $\frac{\alpha}{3}$ goes to infinity under iteration.