Length orthospectrum of convex bodies on flat tori
Résumé
In analogy with the study of Pollicott-Ruelle resonances on negatively curved manifolds, we define anisotropic Sobolev spaces that are well-adapted to the analysis of the geodesic vector field on the flat torus. Among several applications of this functional point of view, we study properties of geodesics that are orthogonal to two convex subsets of the torus (i.e. projection of the boundaries of strictly convex bodies of the Euclidean space). Associated to the set of lengths of such orthogeodesics, we define a geometric Epstein function and prove its meromorphic continuation. We compute its residues in terms of intrinsic volumes of the convex sets. We also prove Poisson-type summation formulas relating the set of lengths of orthogeodesics and the spectrum of magnetic Laplacians.
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