Quasicircles modulo bilipschitz maps

Abstract

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