On the area rate of perturbed Siegel disks

TL;DR

This paper demonstrates that the restrictive conditions on rotation numbers in the construction of quadratic Julia sets with positive area can be removed, enabling the creation of more such sets.

Contribution

It removes a key restriction on rotation numbers in the construction of positive-area quadratic Julia sets, broadening the class of such sets.

Findings

More quadratic Julia sets with positive area can be constructed.

The restrictive condition on rotation numbers is no longer necessary.

The technique used in Buff and Chéritat's work is generalized.

Abstract

In the famous work by Buff and Ch\'eritat constructing quadratic Julia sets with positive area, the control of the shape of perturbed Siegel disks is a key technique. To do it, Buff and Ch\'eritat added a restrictive condition on rotation numbers. In this paper, we show that this restrictive condition can be deleted. As a consequence, more quadratic Julia sets with positive area can be constructed.