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Conformal Geometry and Dynamics

ISSN 1088-4173



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

Author: Jane M. Hawkins
Journal: Conform. Geom. Dyn. 17 (2013), 26-38
MSC (2010): Primary 37F10, 37F45; Secondary 30D05, 30B99
Published electronically: February 14, 2013
MathSciNet review: 3019711
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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 $p$.

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Additional Information

Jane M. Hawkins
Affiliation: Department of Mathematics, University of North Carolina at Chapel Hill, CB #3250, Chapel Hill, North Carolina 27599-3250
MR Author ID: 82840

Keywords: Connected Julia sets, complex dynamics, iterated elliptic functions
Received by editor(s): July 15, 2012
Published electronically: February 14, 2013
Additional Notes: This work was partially funded by a University of North Carolina, University Research Council Grant
Article copyright: © Copyright 2013 American Mathematical Society