Circle Homeomorphisms with Square Summable Diamond Shears

Abstract

We introduce and study the space of homeomorphisms of the circle (up to Möbius 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 Teichmüller space introduced by Penner. We obtain sharp results comparing this new class to the Weil–Petersson class and Hölder classes of circle homeomorphisms. We also express the Weil–Petersson metric tensor and symplectic form in terms of infinitesimal shears and diamond shears.