Ergodic Theory and Dynamical Systems

Research Article

Rational maps whose Fatou components are Jordan domains

Kevin M. Pilgrima1

a1 Mathematics Department, White Hall, Cornell University, Ithaca, NY 14853, USA, (e-mail: pilgrim@math.Cornell.edu)

Abstract

We prove: If f(z) is a critically finite rational map which has exactly two critical points and which is not conjugate to a polynomial, then the boundary of every Fatou component of f is a Jordan curve. If f(z) is a hyperbolic critically finite rational map all of whose postcritical points are periodic, then there exists a cycle of Fatou components whose boundaries are Jordan curves. We give examples of critically finite hyperbolic rational maps f with a Fatou component ω satisfying f(ω) = ω and f|∂ω not topologically conjugate to the dynamics of any polynomial on its Julia set.

(Received January 05 1995)

(Revised September 01 1995)

Footnotes

† This research was supported by a National Need Fellowship and NSF Grant DMS-9301502. Research at MSRI was supported in part by NSF grant no. DMS-9022140.