On postcritical sets of quadratic polynomials with a neutral fixed point

TL;DR

This paper advances the understanding of postcritical sets of quadratic polynomials with neutral fixed points by extending control techniques to the general case, building on prior theories and analytic methods.

Contribution

It provides a unified control framework for postcritical sets of quadratic polynomials with neutral fixed points, generalizing previous results to all rotation numbers.

Findings

Extended control to the general case of rotation numbers.

Built on Inou-Shishikura and Cheraghi's theories.

Utilized pseudo-Siegel disk theory of Dudko and Lyubich.

Abstract

The control of postcritical sets of quadratic polynomials with a neutral fixed point is a main ingredient in the remarkable work of Buff and Ch\'eritat to construct quadratic Julia sets with positive area. Based on the Inou-Shishikura theory, they obtained the control for the case of rotation numbers of bounded high type. Later, Cheraghi developed several elaborate analytic techniques based on Inou-Shishikura's results and obtained the control for the case of rotation numbers of high type. In this paper, based on the pseudo-Siegel disk theory of Dudko and Lyubich, we obtained the control for the general case.