Piecewise quasiconformal dynamical systems of the unit circle

TL;DR

This paper investigates piecewise quasiconformal maps of the unit circle, establishing conditions for conjugacies to extend quasiconformally, and applies these results to classify maps, prove a mating theorem, and analyze parabolic basins.

Contribution

It generalizes previous results on piecewise analytic maps to quasiconformal maps, providing new classification and extension theorems for these dynamical systems.

Findings

Established conditions for quasiconformal and David extensions of conjugacies.

Classified piecewise quasiconformal maps up to quasisymmetric conjugacy.

Proved a conformal mating theorem for Blaschke products.

Abstract

We study piecewise quasiconformal covering maps of the unit circle. We provide sufficient conditions so that a conjugacy between two such dynamical systems has a quasiconformal or David extension to the unit disk. Our main result generalizes the main result of arXiv:2010.11256, which deals with piecewise analytic maps. As applications, we provide a classification of piecewise quasiconformal maps of the circle up to quasisymmetric conjugacy, we prove a general conformal mating theorem for Blaschke products, and we study the quasiconformal geometry of parabolic basins.