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Circle homeomorphisms with square summable diamond shears

We introduce and study `2 spaces of homeomorphisms of the circle (up to Möbius transformations) with respect to two modular coordinates, namely shears and diamond shears along the edges of the Farey tessellation. Shears and diamond shears are related combinatorially, and the latter is closely related to the log Λ-lengths of decorated universal Teichmüller space introduced by R. Penner. We obtain sharp results comparing these new classes 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.