Given an holomorphic function $f:\mathbb{C}\to\mathbb{C}$, I got stuck trying to prove the equality of both Julia sets: $J(f) = J(f^p)$ for every $p\in\mathbb{N}$.
$J(f^p)\subset J(f)$ is easy and I don't have troubles, but I don't know how to prove the reciprocal $J(f)\subset J(f^p)$.
These are the definitions required:
A family of holomorphic maps $\mathcal{F}$ is said to be a normal family if any infinite sequence of elements of $\mathcal{F}$ contains a subsequence which converges uniformly on compact sets of their domain.
Julia set $J(f)$ is the complementary set of the Fatou set $F_f=\{z\text{ such that }\{f^n\}_n\text{ is normal in a neighbourhood of }z\}$.
Thanks in advance!
$J(f)\subset J(f^p)$ is equivalent to $F(f^p)\subset F(f)$.
Let $z_0 \in F(f^p)$ and $U$ a neighborhood of $z_0$ in which the family of iterates $\{ f^{np} | n \in \Bbb N \}$ is normal. Then – since $f$ is uniformly continuous on compact sets – each of the families $$ A_q = \{ f^{np+q} | n \in \Bbb N \} \quad (0 \le q \le p-1) $$ is normal in $U$.
Now let $(f^{n_j})_j$ be a subsequence of the iterates of $f$. Then at least one of the $A_q$ contains infinitely many of the $f^{n_j}$, i.e. there is a subsequence $(f^{n_{j_k}})_k$ which is contained in some $A_q$, and therefore has a convergent subsequence.
This shows that $\{ f^{n} | n \in \Bbb N \}$ is normal in a neighborhood of $z_0$.