Rational Maps Whose Fatou Components Are Jordan Domains

Kevin M. Pilgrim

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arXiv · math/9412205

Rational Maps Whose Fatou Components Are Jordan Domains

Kevin M. Pilgrim

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

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Kevin M. Pilgrim. 1994-12-19. Rational Maps Whose Fatou Components Are Jordan Domains. https://arxiv.org/abs/math/9412205

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