The fixed point of the parabolic renormalization operator

TL;DR

This paper investigates the properties of parabolic renormalization for certain analytic maps, proves the invariance of a specific class under this process, and introduces numerical methods to approximate the fixed point and analyze its stability.

Contribution

It identifies a class of maps invariant under parabolic renormalization, proves the fixed point lies within this class, and develops numerical techniques to approximate the fixed point and its spectral properties.

Findings

The class $ extbf{P}$ is invariant under parabolic renormalization.

The fixed point $f_*$ is contained in $ extbf{P}$.

Numerical estimates of the spectral radius at $f_*$ are provided.

Abstract

We study parabolic renormalization of analytic germs with a simple parabolic point at the origin. We describe a class of maps $\mathbf P$ which admit a maximal analytic extension to a Jordan domain, and whose covering properties have an explicit topological model. We demonstrate that $\mathbf P$ is invariant under parabolic renormalization, and that Inou-Shishikura fixed point $f_*$ lies in $\mathbf P$. We conjecture that successive parabolic renormalizations of every map in $\mathbf P$ converge to $f_*$ at a geometric rate. We further present a numerical method for computing the Taylor's expansion of $f_*$ with a high accuracy. Our approach also allows us to compute the images of the maximal domain of analyticity of $f_*$. Finally, we obtain numerical estimates on the spectral radius of the differential of the parabolic renormalization operator at $f_*$.