RATIONAL MAPS WHOSE FATOU

COMPONENTS ARE JORDAN DOMAINS

Kevin Pilgrim

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 the property that on the closure of a Fatou component Ω satisfying f(Ω) = Ω, f|_{∂Ω} is not topologically conjugate to the dynamics of any polynomial on its Julia set.

Typeset by AMS-TEX