The parabolic and near-parabolic renormalization for a class of polynomial maps and its applications

TL;DR

This paper introduces parabolic and near-parabolic renormalization techniques for polynomial maps with high degrees and parabolic fixed points, demonstrating the existence of Julia sets with positive area and Cremer points, advancing understanding in complex dynamics.

Contribution

It develops new renormalization methods for polynomial maps with degrees over 21 and proves the existence of non-renormalizable maps with positive-area Julia sets, addressing a classical conjecture.

Findings

Existence of polynomial maps with positive-area Julia sets and Cremer points.

Introduction of parabolic and near-parabolic renormalization techniques.

Validation of the Fatou conjecture for degrees greater than 21.

Abstract

For a class of polynomial maps of one variable with a parabolic fixed points and degrees bigger than $21$, the parabolic renormalization is introduced based on Fatou coordinates and horn maps, and a type of maps which are invariant under the parabolic renormalization is also given. For the small perturbation of these kinds of maps, the near-parabolic renormalization is also introduced based on the first return maps defined on the fundamental regions. As an application, we show the existence of non-renormalizable polynomial maps with degrees bigger than $21$ such that the Julia sets have positive Lebesgue measure and Cremer fixed points, this provides a positive answer for the classical Fatou conjecture (the existence of Julia set with positive area) with degrees bigger than $21$.