An optimal Yoccoz inequality for near-parabolic quadratic polynomials

TL;DR

This paper develops a quadratic Pommerenke-Levin-Yoccoz inequality for near-parabolic quadratic polynomials, offering new geometric bounds on the Mandelbrot set's limbs by analyzing external and parameter rays using Lavaurs maps and near-parabolic renormalization.

Contribution

It introduces a novel quadratic inequality for near-parabolic parameters, improving understanding of the Mandelbrot set's geometry near parabolic points.

Findings

Established new bounds on Mandelbrot set limbs.

Derived a quadratic Pommerenke-Levin-Yoccoz inequality.

Analyzed the geometry of external and parameter rays near parabolic points.

Abstract

Using Lavaurs maps and near-parabolic renormalization, we describe the degenerating geometry of external rays for quadratic polynomials when a periodic cycle becomes parabolic. We similarly describe the geometry of parameter rays for the Mandelbrot set near parabolic points. Using this geometric control we establish new bounds on the size of limbs of the Mandelbrot set, providing a quadratic Pommerenke-Levin-Yoccoz inequality in the near-parabolic setting.