Uniformization of gasket Julia sets

TL;DR

This paper characterizes gasket Julia sets of rational maps that can be uniformized by round gaskets, establishing conditions under which such uniformizations are possible and relating them to geometric properties of Fatou components.

Contribution

It provides a complete characterization of when gasket Julia sets can be quasiconformally uniformized by round gaskets, linking geometric conditions to uniformization possibilities.

Findings

Julia sets can be quasiconformally uniformized by round gaskets if and only if they are fat gaskets.

Uniformization by a David homeomorphism occurs if and only if all Fatou components are quasidisks.

Gasket Julia sets and Kleinian limit sets can be locally quasiconformally homeomorphic.

Abstract

The object of the paper is to characterize gasket Julia sets of rational maps that can be uniformized by round gaskets. We restrict to rational maps without critical points on the Julia set. Under these conditions, we prove that a Julia set can be quasiconformally uniformized by a round gasket if and only if it is a fat gasket, i.e., boundaries of Fatou components intersect tangentially. We also prove that a Julia set can be uniformized by a round gasket with a David homeomorphism if and only if every Fatou component is a quasidisk; equivalently, there are no parabolic cycles of multiplicity 2. Our theorem applies to show that gasket Julia sets and limit sets of Kleinian groups can be locally quasiconformally homeomorphic, although globally this is conjectured to be false.