Quasicircles modulo bilipschitz maps

Abstract

We give an explicit construction of all quasicircles, modulo bilipschitz maps. More precisely, we construct a class $\mathcal{S}$ of planar Jordan curves, using a process similar to the construction of the van Koch snowflake curve. These snowflake-like curves are easily seen to be quasicircles. We prove that for every quasicircle $\Gamma$ there is a bilipschitz homeomorphism $f$ of the plane and a snowflake-like curve $S\in \mathcal{S}$ with $\Gamma =f(S)$. In the same fashion we obtain a construction of all bilipschitz-homogeneous Jordan curves, modulo bilipschitz map, and determine all dimension functions occuring for such curves. As a tool we construct a variant of the Konyagin–Volberg uniformly doubling measure on $\Gamma$.