Periodic Beurling-Ahlfors Extensions and Quasisymmetric Rigidity of 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 ${\mathbb{C}}^{\ast }$-square carpets is discussed.