Circle homeomorphisms and shears

TL;DR

This paper introduces a new way to parameterize homeomorphisms and related maps of the unit circle using shear coordinates linked to the Farey tessellation, enhancing understanding of circle homeomorphisms.

Contribution

It provides novel parameterizations of circle homeomorphisms, quasisymmetric, and symmetric maps via shear coordinates for the Farey tessellation, connecting geometric structures with map classifications.

Findings

Parameterizations of circle homeomorphisms using shear coordinates

Connections between shear coordinates and quasisymmetric maps

Enhanced understanding of symmetric maps on the circle

Abstract

We give parameterizations of homeomorphisms, quasisymmetric maps and symmetric maps of the unit circle in terms of shear coordinates for the Farey tesselation.