Structure of the McMullen domain in the parameter planes for rational maps

Abstract

We show that, for the family of functions F�(z) = z n +�/z n where n � 3 and � 2 C, there is a unique McMullen domain in parameter space. A McMullen domain is a region where the Julia set of Fis homeomorphic to a Cantor set of circles. We also prove that this McMullen domain is a simply connected region in the plane that is bounded by a simple closed curve.