Koebe uniformization for infinitely connected attracting Fatou domains

TL;DR

This paper establishes that for geometrically finite rational maps, their infinitely connected attracting Fatou domains can be conformally mapped to circle domains, extending uniformization results to complex boundary structures.

Contribution

It proves that geometrically finite rational maps have infinitely connected attracting Fatou domains conformally equivalent to circle domains, addressing boundary complexity issues.

Findings

Infinitely connected attracting Fatou domains are conformally homeomorphic to circle domains under geometric finiteness.

Existing uniformization results do not apply to complex boundary behaviors, but this work overcomes that limitation.

The paper advances understanding of Fatou domain structures in rational dynamics.

Abstract

This paper works on the structure of infinitely connected Fatou damains of rational maps in terms of Koebe uniformization. Due to the complicated boundary behavior, the existing uniformization results are failed to apply in general. We proved that if the rational map is geometrically finite, then its infinitely connected attracting Fatou damain is conformally homeomorphic to a circle domain.