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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The McMullen domain: Rings around the boundary
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by Robert L. Devaney and Sebastian M. Marotta PDF
Trans. Amer. Math. Soc. 359 (2007), 3251-3273 Request permission

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
  • MR Author ID: 57240
  • Sebastian M. Marotta
  • Affiliation: Department of Mathematics, Boston University, 111 Cummington Street, Boston, Massachusetts 02215
  • Received by editor(s): May 5, 2005
  • Published electronically: February 13, 2007
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 3251-3273
  • MSC (2000): Primary 37F10; Secondary 37F45
  • DOI: https://doi.org/10.1090/S0002-9947-07-04137-2
  • MathSciNet review: 2299454