Continuity of the barycentric extension of circle diffeomorphisms of H\"older continuous derivatives

TL;DR

This paper proves the continuity of the barycentric extension for circle diffeomorphisms with H"older continuous derivatives, ensuring a stable extension process in the space of Beltrami coefficients.

Contribution

It establishes the continuity of the barycentric extension for a new class of circle diffeomorphisms with H"older derivatives, extending previous results.

Findings

Continuity of barycentric extension in the H"older continuous derivatives setting

Extension operation is continuous with respect to Beltrami coefficients

Provides a new understanding of the stability of conformal extensions

Abstract

The barycentric extension due to Douady and Earle gives a conformally natural extension of a quasisymmetric automorphism of the circle to a quasiconformal automorphism of the unit disk. We consider such extensions for circle diffeomorphisms of H\"older continuous derivatives and show that this operation is continuous with respect to an appropriate topology for the space of the corresponding Beltrami coefficients.