Volume 7 (2024)
Nakhlé, Elie;  Sabourau, Stéphane
Deforming a Finsler metric on the two-torus to a flat Finsler metric with conjugate geodesic flows
Annales Henri Lebesgue, Volume 7 (2024), pp. 1131-1174.

Metadata

DOI10.5802/ahl.217
Keywords Finsler metrics, dynamical systems, geodesic flow, conjugate flows, conjugate points, integral geometry, Crofton formula, Heber foliation, curve shortening flow

Abstract

We show that the space of (reversible) Finsler metrics on the two-torus 𝕋 2 whose geodesic flow is conjugate to the geodesic flow of a flat Finsler metric strongly deformation retracts to the space of flat Finsler metrics with respect to the uniform convergence topology. Along the proof, we also show that two Finsler metrics on 𝕋 2 without conjugate points, whose Heber foliations are smooth and with the same marked length spectrum, have conjugate geodesic flows.


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