Rational Maps Whose Fatou Components Are Jordan Domains

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 $\Omega$ satisfying $f(\Omega )=\Omega$, $f|_{\bdry Ω}$ is not topologically conjugate to the dynamics of any polynomial on its Julia set.

Separate uu-encoded "tar" file of figures sent also. Uses Latex2.09 and geompsfi.sty

References in corpus (3)

• Dynamics in one complex variable: introductory lectures
• Remarks on quadratic rational maps
• Surgery on postcritically finite rational maps by blowing up an arc