We give an explicit construction of all quasicircles, modulo bilipschitz maps. More precisely, we construct a class 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 there is a bilipschitz homeomorphism of the plane and a snowflake-like curve \( S \in \mathcal S \) with . 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 .
Cite this article
Steffen Rohde, Quasicircles modulo bilipschitz maps. Rev. Mat. Iberoam. 17 (2001), no. 3, pp. 643–659DOI 10.4171/RMI/307