Bounded turning circles are weak-quasicircles

TL;DR

This paper establishes an equivalence between bounded turning Jordan curves and weak-quasisymmetric images of the circle, providing a new characterization of such curves in metric geometry.

Contribution

It proves that bounded turning Jordan curves are precisely the images of the circle under weak-quasisymmetric homeomorphisms, linking geometric and analytic properties.

Findings

Bounded turning curves are characterized by weak-quasisymmetric images.

The paper provides a new geometric-analytic equivalence for Jordan curves.

This characterization aids in understanding the structure of metric Jordan curves.

Abstract

We show that a metric Jordan curve $\Gamma$ is \emph{bounded turning} if and only if there exists a \emph{weak-quasisymmetric} homeomorphism $\phi\colon \mathsf{S}^1 \to \Gamma$.