Prove that non-attracting periodic orbits of $x \mapsto x^2 + c$ are in the Julia set

Question

Let $c$ be complex number and let \begin{align} f_c : \mathbb{C} &\to \mathbb{C} \\ x &\mapsto x^2 + c \end{align} be the quadratic map. It is to be shown that if $O$ is a periodic orbit that is not an attractor, then $O$ must lie in the Julia set of $f_c$.

I am not sure how to proceed.

Answer 1

This is not true. If $z$ is a periodic orbit of period $p$, its multiplier is $m:=(f_c^p)'(z)$. We have the following (rough) classification:

• attracting if $|m|<1$

• repelling if $|m|>1$

• rationnally neutral if $m$ is a root of unity

• irrationnally neutral if $m=e^{2i\pi t}$, $t \in \mathbb R - \mathbb Q$.

Repelling and rationnally neutral periodic points are always in the Julia set, but some irrationnally neutral periodic points are in the Fatou set (if and only if they are linearizable).

This classification of periodic orbits is discussed in any textbook in complex dynamics, eg the book of Milnor or Carleson-Gamelin. Determining which values of $m$ give irrationnally neutral periodic points in the Julia set is a very complicated question, far beyond homework, and indeed not completely solved yet.