Length orthospectrum of convex bodies on flat tori
Copyright policy )Length orthospectrum of convex bodies on flat tori
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.
- French National Centre for Scientific Research France
- University of Paris France
- Sorbonne Paris Cité France
- Sorbonne University France
- UNIVERSITE PARIS DESCARTES France
Mathematics - Differential Geometry, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], [MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS], FOS: Physical sciences, Mathematical Physics (math-ph), Dynamical Systems (math.DS), 510, Mathematics - Spectral Theory, Mathematics - Analysis of PDEs, Differential Geometry (math.DG), [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], [MATH.MATH-SP] Mathematics [math]/Spectral Theory [math.SP], FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph], Mathematics - Dynamical Systems, [MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG], Spectral Theory (math.SP), Mathematical Physics, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP], Analysis of PDEs (math.AP)
Mathematics - Differential Geometry, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], [MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS], FOS: Physical sciences, Mathematical Physics (math-ph), Dynamical Systems (math.DS), 510, Mathematics - Spectral Theory, Mathematics - Analysis of PDEs, Differential Geometry (math.DG), [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], [MATH.MATH-SP] Mathematics [math]/Spectral Theory [math.SP], FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph], Mathematics - Dynamical Systems, [MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG], Spectral Theory (math.SP), Mathematical Physics, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP], Analysis of PDEs (math.AP)
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