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Conformal Geometry and Dynamics

Published by the American Mathematical Society since 1997, the purpose of this electronic-only journal is to provide a forum for mathematical work in related fields broadly described as conformal geometry and dynamics. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.49.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


The McMullen domain: Satellite Mandelbrot sets and Sierpinski holes
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by Robert L. Devaney
Conform. Geom. Dyn. 11 (2007), 164-190
Published electronically: September 20, 2007


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 $k$.

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Bibliographic 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:
  • MathSciNet review: 2346215