Rings Around Sierpinski Holes

Abstract

For the family of rational maps [Formula: see text] where [Formula: see text] it is known that there are infinitely many Mandelpiński necklaces [Formula: see text] with [Formula: see text] around the McMullen domain surrounding the origin in the parameter [Formula: see text]-plane. In this paper, we prove the existence of infinitely many these rings with a number of [Formula: see text] for fixed [Formula: see text] outside the Mandelpiński necklace [Formula: see text]. The ring [Formula: see text] is a simple closed curve meeting [Formula: see text] at [Formula: see text] points, such that it passes through exactly [Formula: see text] centers of Sierpinski holes and [Formula: see text] superstable parameter values. For each [Formula: see text], [Formula: see text] passes through precisely alternating [Formula: see text] superstable parameter values and the same number of centers of Sierpiński holes. There exist [Formula: see text] disjoint rings [Formula: see text] not meeting [Formula: see text] and surrounding the centers of Sierpiński holes lying on [Formula: see text], in the exterior and interior of a curve [Formula: see text], respectively. The number of such rings [Formula: see text] for fixed [Formula: see text] is [Formula: see text].