The fixed point of the parabolic renormalization operator

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 $f_{\ast }$ lies in $\mathbf{P}$. We conjecture that successive parabolic renormalizations of every map in $\mathbf{P}$ converge to $f_{\ast }$ at a geometric rate. We further present a numerical method for computing the Taylor's expansion of $f_{\ast }$ with a high accuracy. Our approach also allows us to compute the images of the maximal domain of analyticity of $f_{\ast }$. Finally, we obtain numerical estimates on the spectral radius of the differential of the parabolic renormalization operator at $f_{\ast }$.

Improvements to exposition, and various typos fixed

Cited by in corpus (2)

• A Proof of the Marmi-Moussa-Yoccoz conjecture for rotation numbers of high type
• The resurgent character of the Fatou coordinates of a simple parabolic germ