Idempotent entire complex function problem Problem statement
Find all the entire functions $f:\mathbb C \to \mathbb C$, that satisfy $f(f(z))=f(z)$ for all $z \in \mathbb C$.
I have no idea how to attack this problem, I would appreciate hints and suggestions.

Hints: (Just throwing some things out there that you can use. Quite a few ways to solve it)
(1) Think about the identity function
(2) Identity theorem
(3) Think about the range of $f$, also consider the following: If two continuous functions $f, g : X → Y$ into a Hausdorff space $Y$ agree on a dense subset of $X$ then they agree on all of $X$. 
(4) If an entire function is bounded then it has to be constant, see Liouville's theorem.
