The McMullen domain: Satellite Mandelbrot sets and Sierpinski holes
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 Conform. Geom. Dyn. 11 (2007), 164190 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:

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 $k \rightarrow \infty$.

The curve ${\mathcal S}^k$ meets the centers of $\tau _k^n$ Sierpinski holes, each with escape time $k+2$ where \[ \tau _{k}^n = (n2) n^{k1} + 1. \]

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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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), 164190
 MSC (2000): Primary 37F45; Secondary 37F20
 DOI: https://doi.org/10.1090/S108841730700166X
 MathSciNet review: 2346215