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This note presents some of the results obtained in arXiv:2207.05410 and ithas beenthe object of a talk of the second author during the Journ\'ees"\'Equations auxD\'eriv\'ees Partielles" (Obernai, june 2022). We studyproperties of geodesics that are orthogonal to two convex subsets of the flattorus. We discuss meromorphic properties of a geometric Epstein zeta functionassociated to the set of lengths of such orthogeodesics. We also define theassociated length distribution and discuss singularities of its Fouriertransform. Our analysis relies on a fine study of the dynamical correlationfunction of the geodesic flow on the torus and the definition of anisotropicSobolev spaces that are well-adapted to this integrable dynamics.

Length orthospectrum and the correlation function on flat tori