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
DOI:
https://doi.org/10.1090/S1088-4173-2013-00252-X
Published electronically:
February 14, 2013
MathSciNet review:
3019711
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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
Email:
jmh@math.unc.edu
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