The Mcmullen Domain: Satellite Mandelbrot Sets and Sierpinski Holes

In this paper we describe some features of the parameter planes for the families of rational maps given by Fλ(z) = z n + λ/zn 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 Sk for k = 1, 2, . . . having the property that: (1) Each curve Sk surrounds the McMullen domain as well as Sk+1, and the Sk accumulate on the boundary of the McMullen domain as k → ∞. (2) The curve Sk meets the centers of τn k Sierpinski holes, each with escape time k + 2 where τ k = (n− 2)nk−1 + 1. (3) The curve Sk also passes through τn k parameter values which are centers of the main cardioids of baby Mandelbrot sets with base period k.