Circle homeomorphisms with square summable diamond shears

TL;DR

This paper introduces a new class of circle homeomorphisms characterized by square-summable diamond shears, connecting geometric, combinatorial, and Teichmüller theory concepts, and compares it to established classes like Weil-Petersson and Hölder classes.

Contribution

It defines and studies the space of circle homeomorphisms with square-summable diamond shears, relating it to Teichmüller theory and providing new metric and symplectic formulations.

Findings

Characterization of the new class via $ ext{ell}^2$ diamond shears.

Comparison with Weil-Petersson and Hölder classes.

Expression of Weil-Petersson metric and symplectic form in shear coordinates.

Abstract

We introduce and the study the space of homeomorphisms of the circle (up to M\"obius transformations) which are in $\ell^2$ with respect to modular coordinates called diamond shears along the edges of the Farey tessellation. Diamond shears are related combinatorially to shear coordinates, and are also closely related to the $\log \Lambda$-lengths of decorated Teichm\"uller space introduced by Penner. We obtain sharp results comparing this new class to the Weil-Petersson class and H\"older classes of circle homeomorphisms. We also express the Weil-Petersson metric tensor and symplectic form in terms of infinitesimal shears and diamond shears.