Revista Matemática Iberoamericana
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Published online: 2001-12-31
Quasicircles modulo bilipschitz mapsSteffen Rohde (1) University of Washington, Seattle, USA
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 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 $\Gamma$.
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Rohde Steffen: Quasicircles modulo bilipschitz maps. Rev. Mat. Iberoam. 17 (2001), 643-659. doi: 10.4171/RMI/307