Periodic Beurling-Ahlfors Extensions and Quasisymmetric Rigidity of Carpets

TL;DR

This paper proves periodic quasiconformal extension theorems for quasisymmetric maps on carpets and establishes rigidity results, showing such maps are identities under certain fixed point conditions, advancing understanding of geometric structures in complex analysis.

Contribution

The paper introduces new periodic extension theorems for quasisymmetric maps on carpets and demonstrates rigidity results, showing these maps are identities when fixing specific points or structures.

Findings

Quasiconformal extension theorems for periodic quasisymmetric maps

Rigidity results for maps fixing boundary points

Applications to square and $ ext{C}^*$-square carpets

Abstract

We establish periodic quasiconformal extension theorems for periodic orientation-preserving quasisymmetric self homeomorphisms of quasicircles or quasi-round carpets. As applications, we prove that, if $f$ is a periodic orientation-preserving quasisymmetric self homeomorphism of a quasi-round carpet $S$ of measure zero in $\mathbb{C}$, which has a fixed point in the outer peripheral circle of $S$, then $f$ is the identity on $S$. Moreover, we prove that, if $f$ is a quasisymmetric self homeomorphism of a square carpet $S$ of measure zero in a rectangle ring, which fixes each of the four vertices of the outer peripheral circle of $S$, then $f$ is the identity on $S$. An analogous rigidity problem for the $\mathbb{C}^*$-square carpets is discussed.