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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The McMullen domain: Rings around the boundary


Authors: Robert L. Devaney and Sebastian M. Marotta
Journal: Trans. Amer. Math. Soc. 359 (2007), 3251-3273
MSC (2000): Primary 37F10; Secondary 37F45
Published electronically: February 13, 2007
MathSciNet review: 2299454
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Abstract: In this paper we show that there are infinitely many rings $ {\mathcal S}^k, k \geq 1$, around the McMullen domain in the parameter plane for the family of complex rational maps of the form $ z^n + \lambda/z^n$ where $ \lambda \in \mathbb{C}$ and $ n \geq 3$. These rings converge to the boundary of the McMullen domain as $ k \rightarrow \infty$. The rings $ {\mathcal S}^k$ contain $ (n-2)n^{k-1} + 1$ parameter values that lie at the center of Sierpinski holes. That is, these parameters lie at the center of an open set in the parameter plane in which all of the corresponding maps have Julia sets that are Sierpinski curves. The rings also contain the same number of superstable parameter values, i.e., parameter values for which one of the critical points is periodic of period either $ k$ or $ 2k$.


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Additional Information

Robert L. Devaney
Affiliation: Department of Mathematics, Boston University, 111 Cummington Street, Boston, Massachusetts 02215

Sebastian M. Marotta
Affiliation: Department of Mathematics, Boston University, 111 Cummington Street, Boston, Massachusetts 02215

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04137-2
PII: S 0002-9947(07)04137-2
Received by editor(s): May 5, 2005
Published electronically: February 13, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.