The McMullen domain: Satellite Mandelbrot sets and Sierpinski holes
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 by Robert L. Devaney
 Conform. Geom. Dyn. 11 (2007), 164190
 DOI: https://doi.org/10.1090/S108841730700166X
 Published electronically: September 20, 2007
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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 $k$.
References
 Paul Blanchard, Robert L. Devaney, Daniel M. Look, Pradipta Seal, and Yakov Shapiro, Sierpinskicurve Julia sets and singular perturbations of complex polynomials, Ergodic Theory Dynam. Systems 25 (2005), no. 4, 1047–1055. MR 2158396, DOI 10.1017/S0143385704000380
 Robert L. Devaney, Baby Mandelbrot sets adorned with halos in families of rational maps, Complex dynamics, Contemp. Math., vol. 396, Amer. Math. Soc., Providence, RI, 2006, pp. 37–50. MR 2209085, DOI 10.1090/conm/396/07392
 Robert L. Devaney, Structure of the McMullen domain in the parameter planes for rational maps, Fund. Math. 185 (2005), no. 3, 267–285. MR 2161407, DOI 10.4064/fm18535
 Robert L. Devaney and Sebastian M. Marotta, The McMullen domain: rings around the boundary, Trans. Amer. Math. Soc. 359 (2007), no. 7, 3251–3273. MR 2299454, DOI 10.1090/S0002994707041372
 Robert L. Devaney and Daniel M. Look, A criterion for Sierpinski curve Julia sets, Topology Proc. 30 (2006), no. 1, 163–179. Spring Topology and Dynamical Systems Conference. MR 2280665
 Robert L. Devaney, Daniel M. Look, and David Uminsky, The escape trichotomy for singularly perturbed rational maps, Indiana Univ. Math. J. 54 (2005), no. 6, 1621–1634. MR 2189680, DOI 10.1512/iumj.2005.54.2615
 Douady, A. and Hubbard, J., Etude Dynamique des Polynômes Complexes. Publ. Math. D’Orsay (1984).
 Adrien Douady and John Hamal Hubbard, On the dynamics of polynomiallike mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 2, 287–343. MR 816367
 R. Mañé, P. Sad, and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193–217. MR 732343
 Curt McMullen, Automorphisms of rational maps, Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 10, Springer, New York, 1988, pp. 31–60. MR 955807, DOI 10.1007/9781461396024_{3}
 Curtis T. McMullen, The classification of conformal dynamical systems, Current developments in mathematics, 1995 (Cambridge, MA), Int. Press, Cambridge, MA, 1994, pp. 323–360. MR 1474980
 John Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309
 John Milnor, Geometry and dynamics of quadratic rational maps, Experiment. Math. 2 (1993), no. 1, 37–83. With an appendix by the author and Lei Tan. MR 1246482
 Carsten Lunde Petersen and Gustav Ryd, Convergence of rational rays in parameter spaces, The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, Cambridge, 2000, pp. 161–172. MR 1765088
 Roesch, P., On Capture Zones for the Family $f_\lambda (z) = z^2 + \lambda /z^2$. In Dynamics on the Riemann Sphere, European Mathematical Society, (2006), 121130.
 G. T. Whyburn, Topological characterization of the Sierpiński curve, Fund. Math. 45 (1958), 320–324. MR 99638, DOI 10.4064/fm451320324
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), 164190
 MSC (2000): Primary 37F45; Secondary 37F20
 DOI: https://doi.org/10.1090/S108841730700166X
 MathSciNet review: 2346215