The McMullen domain: Rings around the boundary

Abstract

In this paper we show that there are infinitely many rings S k , k ≥ 1 {\mathcal S}^k, k \geq 1 , around the McMullen domain in the parameter plane for the family of complex rational maps of the form z n + λ / z n z^n + \lambda /z^n where λ ∈ C \lambda \in \mathbb {C} and n ≥ 3 n \geq 3 . These rings converge to the boundary of the McMullen domain as k → ∞ k \rightarrow \infty . The rings S k {\mathcal S}^k contain ( n − 2 ) n k − 1 + 1 (n-2)n^{k-1} + 1 parameter values that lie at the center of Sierpinski holes. That is, these parameters lie at the center of an open set in the parameter plane in which all of the corresponding maps have Julia sets that are Sierpinski curves. The rings also contain the same number of superstable parameter values, i.e., parameter values for which one of the critical points is periodic of period either k k or 2 k 2k .