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

Volume 185 / 2005

Robert L. Devaney Fundamenta Mathematicae 185 (2005), 267-285 MSC: 37F10, 37F45. DOI: 10.4064/fm185-3-5

Abstract

We show that, for the family of functions $ F_\lambda(z) = z^n + {\lambda\over z^n} $ where $n \geq 3$ and $\lambda \in {\mathbb C}$, there is a unique McMullen domain in parameter space. A McMullen domain is a region where the Julia set of $F_\lambda$ is 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.

Authors

  • Robert L. DevaneyBoston University
    Boston, MA 02215, U.S.A.
    e-mail

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