Let $f(z)$ be an entire function and let $J(f)$ be the boundary of the filled Julia set of $f$.
Conjecture
If $J(f)$ is an analytic Jordan curve then $f(z) = g.inv( g(z)^n )$ where $n$ is a positive integer , $g.inv$ is the (functional) inverse of $g$ and $g$ or $g.inv$ are entire.
Example : $ f(z) = z^2 , J(f) $ is the unit circle.