All bounded type Siegel disks of rational maps are quasi-disks

TL;DR

This paper proves that all bounded type Siegel disks in rational maps are quasi-disks with boundary critical points, confirming the Douady-Sullivan conjecture for this class of Siegel disks.

Contribution

It establishes that every bounded type Siegel disk in a rational map is a quasi-disk with at least one boundary critical point, verifying a longstanding conjecture.

Findings

Bounded type Siegel disks are quasi-disks.

At least one critical point lies on the boundary of these disks.

The Douady-Sullivan conjecture is confirmed for bounded type Siegel disks.

Abstract

We prove that every bounded type Siegel disk of a rational map must be a quasi-disk with at least one critical point on its boundary. This verifies Douady-Sullivan conjecture for bounded type Siegel disks.