Quasicircles and bounded turning circles modulo bi-Lipschitz maps (Q446404)

Summary: 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 \textit{S. Rohde} [Rev. Mat. Iberoam. 17, No. 3, 643--659 (2001; Zbl 1003.30013)]. 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.