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Conformal Geometry and Dynamics

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

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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The McMullen domain: Satellite Mandelbrot sets and Sierpinski holes
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by Robert L. Devaney PDF
Conform. Geom. Dyn. 11 (2007), 164-190 Request permission

Abstract:

In this paper we describe some features of the parameter planes for the families of rational maps given by $F_\lambda (z) = z^n + \lambda /z^n$ where $n \geq 3, \lambda \in \mathbb {C}$. We assume $n \geq 3$ since, in this case, there is a McMullen domain surrounding the origin in the $\lambda$-plane. This is a region where the corresponding Julia sets are Cantor sets of concentric simple closed curves. We prove here that the McMullen domain in the parameter plane is surrounded by infinitely many simple closed curves ${\mathcal S}^k$ for $k = 1,2,\ldots$ having the property that:

  1. Each curve ${\mathcal S}^k$ surrounds the McMullen domain as well as ${\mathcal S}^{k+1}$, and the ${\mathcal S}^k$ accumulate on the boundary of the McMullen domain as $k \rightarrow \infty$.

  2. The curve ${\mathcal S}^k$ meets the centers of $\tau _k^n$ Sierpinski holes, each with escape time $k+2$ where \[ \tau _{k}^n = (n-2) n^{k-1} + 1. \]

  3. The curve ${\mathcal S}^k$ also passes through $\tau _k^n$ parameter values which are centers of the main cardioids of baby Mandelbrot sets with base period $k$.

References
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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
  • Received by editor(s): July 11, 2006
  • Published electronically: September 20, 2007
  • © Copyright 2007 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 11 (2007), 164-190
  • MSC (2000): Primary 37F45; Secondary 37F20
  • DOI: https://doi.org/10.1090/S1088-4173-07-00166-X
  • MathSciNet review: 2346215