Quasicircles modulo bilipschitz maps

  • Steffen Rohde

    University of Washington, Seattle, USA


We give an explicit construction of all quasicircles, modulo bilipschitz maps. More precisely, we construct a class S\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 Γ\Gamma there is a bilipschitz homeomorphism ff of the plane and a snowflake-like curve S \in \mathcal S with Γ=f(S)\Gamma = 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\Gamma. 

Cite this article

Steffen Rohde, Quasicircles modulo bilipschitz maps. Rev. Mat. Iberoam. 17 (2001), no. 3, pp. 643–659

DOI 10.4171/RMI/307