Proof of a folklore Julia set connectedness theorem and connections with elliptic functions

Abstract:

We prove the following theorem about Julia sets of the maps \[ f_{n,p,\gamma }(z)= z^n + \frac {\gamma }{z^p}, \] for integers $n,p \geq 2$, $\gamma \in \mathbb {C}$ by using techniques developed for the Weierstrass elliptic $\wp$ function and adapted to this setting.

Folklore connectedness theorem: If $f_{n,p,\gamma }$ has a bounded critical orbit, then $J(f_{n,p,\gamma })$ is connected.

This is related to connectivity results by the author and others about $J(\wp )$, where $\wp$ denotes the Weierstrass elliptic $\wp$ function, especially where the period lattice has some symmetry. We illustrate several further connections between the dynamics of some specific elliptic functions and the family $f_{n,p,\gamma }$ for some values of $n$ and $p$.