On quadratic Siegel disks with a class of unbounded type rotation numbers

TL;DR

This paper investigates quadratic polynomials with Siegel disks having unbounded type rotation numbers, proving boundary points are Lebesgue density points of Julia sets, and explores models for quasiconformal surgery related to these disks.

Contribution

It generalizes McMullen's result to unbounded type rotation numbers and analyzes the structure of Siegel disks and their boundary points in quadratic polynomials.

Findings

Boundary points of Siegel disks are Lebesgue density points of Julia sets.

Generalization of McMullen's result to unbounded type rotation numbers.

Critical point is a measurable deep point of the filled-in Julia set.

Abstract

In this paper we explore a class of quadratic polynomials having Siegel disks with unbounded type rotation numbers. We prove that any boundary point of Siegel disks of these polynomials is a Lebesgue density point of their filled-in Julia sets, which generalizes the corresponding result of McMullen for bounded type rotation numbers. As an application, this result can help us construct more quadratic Julia sets with positive area. Moreover, we also explore the canonical candidate model for quasiconformal surgery of quadratic polynomials with Siegel disks. We prove that for any irrational rotation number, any boundary point of ``Siegel disk'' of the canonical candidate model is a Lebesgue density point of its ``filled-in Julia set'', in particular the critical point $1$ is a measurable deep point of the ``filled-in Julia set''.