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Conformal Geometry and Dynamics
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
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove the following theorem about Julia sets of the maps

$\displaystyle 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
Email: jmh@math.unc.edu

DOI: http://dx.doi.org/10.1090/S1088-4173-2013-00252-X
PII: S 1088-4173(2013)00252-X
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