Quasicircles and Bounded Turning Circles Modulo bi-Lipschitz Maps

TL;DR

This paper classifies all snowflake-type metric quasicircles and bounded turning circles up to bi-Lipschitz equivalence, extending known results and providing new insights into their geometric structure.

Contribution

It constructs a comprehensive catalog of metric quasicircles and bounded turning circles up to bi-Lipschitz maps, generalizing Rohde's result to a broader class of metric spaces.

Findings

All metric quasicircles are classified up to bi-Lipschitz equivalence.

A metric quasicircle with Assouad dimension less than two is bi-Lipschitz equivalent to a planar quasicircle.

The construction applies to non-doubling bounded turning metric circles.

Abstract

We construct a catalog, of snowflake type metric circles, that describes all metric quasicircles up to \bl\ 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.