Length orthospectrum and the correlation function on flat tori

TL;DR

This paper investigates the properties of orthogonal geodesics on flat tori, exploring the associated Epstein zeta function, length distribution, and their Fourier transform singularities, using dynamical systems and anisotropic Sobolev spaces.

Contribution

It introduces a detailed analysis of orthogeodesics on flat tori, including the meromorphic properties of a related Epstein zeta function and the singularities of the Fourier transform of the length distribution.

Findings

Meromorphic properties of the geometric Epstein zeta function established.

Analysis of singularities in the Fourier transform of the length distribution.

Application of anisotropic Sobolev spaces to integrable geodesic dynamics.

Abstract

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