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Article Dans Une Revue Cambridge Journal of Mathematics Année : 2023

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 associated with any translation invariant Finsler metric on the torus $\mathbb{T}^d$. Among several applications of this functional point of view, we study properties of geodesics that are orthogonal to two convex subsets of $\mathbb{T}^d$ (i.e. projection of the boundaries of strictly convex bodies of $\mathbb{R}^d$). Associated with 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 formulae relating the set of lengths of orthogeodesics and the spectrum of magnetic Laplacians.
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Dates et versions

hal-03719958 , version 1 (11-07-2022)
hal-03719958 , version 2 (05-11-2023)

Identifiants

Citer

Nguyen Viet Dang, Matthieu Léautaud, Gabriel Rivière. Length orthospectrum of convex bodies on flat tori. Cambridge Journal of Mathematics, 2023, 11 (4), pp.917-1043. ⟨10.4310/CJM.2023.v11.n4.a3⟩. ⟨hal-03719958v2⟩
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