Rigidity of J-rotational rational maps and critical quasicircle maps

TL;DR

This paper establishes rigidity results for holomorphic dynamical systems with rotation quasicircles, extending known rigidity theorems and demonstrating universality and exponential convergence in a broader class of maps.

Contribution

It extends rigidity results to critical quasicircle maps with imbalanced criticalities and proves universality and exponential convergence of renormalization.

Findings

Absence of line fields on Julia sets of certain rational maps.

Rigidity of critical quasicircle maps with bounded type rotation number.

Exponential convergence of renormalization towards a horseshoe attractor.

Abstract

We present a number of rigidity results concerning holomorphic dynamical systems admitting rotation quasicircles. Firstly, we show the absence of line fields on the Julia set of any rational map that is geometrically finite away from a number of rotation quasicircles with bounded type rotation number. As an application, we prove combinatorial rigidity associated to the problem of degeneration of Herman rings of the simplest configuration. Secondly, we extend a result of de Faria and de Melo on the $C^{1+\alpha}$ rigidity of critical circle maps with bounded type rotation number to a larger class of dynamical objects, namely critical quasicircle maps. Unlike critical circle maps, critical quasicircle maps may have imbalanced inner and outer criticalities. As a consequence, we prove dynamical universality and exponential convergence of renormalization towards a horseshoe attractor.