Rational Maps Whose Fatou Components Are Jordan Domains

Details Rational Maps Whose Fatou Components Are Jordan Domains Rational Maps Whose Fatou Components Are Jordan Domains

1994-12-19

arxiv.org/abs/math/9412205v1

Mathematics

Dynamical Systems

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

Scientific paper

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

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