Minimizing entropy for translation surfaces

Abstract

In this note we consider the entropy [5] of unit area translation surfaces in the SL(2, R) orbits of square tiled surfaces that are the union of squares, where the singularities occur at the vertices and the singularities have a common cone angle. We show that the entropy over such orbits is minimized at those surfaces tiled by equilateral triangles where the singularities occur precisely at the vertices.The authors would like to thank L.Bétermin for suggesting the connection with his work [2]. The second author was supported by ERC Grant 833802-Resonances and EPSRC grant EP/T001674/1.1 This avoids the complication of accounting for cylinders of uncountably many parallel geodesics. Alternatively, we could account for these by counting only their free homotopy classes, but then their polynomial growth does not affect the definition of the entropy.