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The McMullen domain: Satellite Mandelbrot sets and Sierpinski holes

2007
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Conformal Geometry and Dynamics
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In this paper we describe some features of the parameter planes for the families of rational maps given by F λ (z) = z n + λ/z n where n ≥ 3, λ ∈ C. We assume n ≥ 3 since, in this case, there is a McMullen domain surrounding the origin in the λ-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 S k for k = 1, 2, . . .

doi:10.1090/s1088-4173-07-00166-x
fatcat:xckrqbl7jng6nllrhvz3hxtnd4