Hyperbolicity of renormalization of critical quasicircle maps

TL;DR

This paper extends hyperbolicity results of renormalization from critical circle maps to a broader class of critical quasicircle maps, introducing a new renormalization operator with a hyperbolic fixed point.

Contribution

It develops a compact analytic renormalization operator for critical quasicircle maps, demonstrating hyperbolicity and describing its stable manifold structure.

Findings

Established hyperbolicity of the Corona Renormalization fixed point.

Identified the stable manifold as critical quasicircle maps with fixed criticality and rotation number.

Extended renormalization theory to maps with distinct inner and outer criticalities.

Abstract

There is a well developed renormalization theory of real analytic critical circle maps by de Faria, de Melo, and Yampolsky. In this paper, we extend Yampolsky's result on hyperbolicity of renormalization periodic points to a larger class of dynamical objects, namely critical quasicircle maps, i.e. analytic self homeomorphisms of a quasicircle with a single critical point. Unlike critical circle maps, the inner and outer criticalities of critical quasicircle maps can be distinct. We develop a compact analytic renormalization operator called Corona Renormalization with a hyperbolic fixed point whose stable manifold has codimension one and consists of critical quasicircle maps of the same criticality and periodic type rotation number. Our proof is an adaptation of Pacman Renormalization Theory for Siegel disks as well as rigidity results on the escaping dynamics of transcendental entire functions.