We construct a catalog, of snowflake type metric circles, that describes all metric quasicircles up to bi-Lipschitz equivalence. This is a metric space analog of a result due to Rohde. Our construction also works for all bounded turning metric circles; these need not be doubling. As a byproduct, we show that a metric quasicircle with Assouad dimension strictly less than two is bi-Lipschitz equivalent to a planar quasicircle.
Cite this article
David A. Herron, Daniel Meyer, Quasicircles and bounded turning circles modulo bi-Lipschitz maps. Rev. Mat. Iberoam. 28 (2012), no. 3, pp. 603–630DOI 10.4171/RMI/687