Deforming a Finsler metric on the two-torus to a flat Finsler metric with conjugate geodesic flows

Nakhlé, Elie; Sabourau, Stéphane

Annales Henri Lebesgue, Tome 7 (2024), pp. 1131-1174

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.

Nous montrons que l’espace des métriques de Finsler (réversibles) sur le tore bidimensionnel ${𝕋}^2$, dont le flot géodésique est conjugué au flot géodésique d’une métrique de Finsler plate se rétracte par déformation forte sur l’espace des métriques de Finsler plates par rapport à la topologie de la convergence uniforme. Au cours de la preuve, nous montrons également que deux métriques de Finsler sur ${𝕋}^2$ sans points conjugués, dont les feuilletages d’Heber sont lisses et ont le même spectre marqué des longueurs, ont des flots géodésiques conjugués.

Classification : 37J39, 53C22, 53C65
Keywords: Finsler metrics, dynamical systems, geodesic flow, conjugate flows, conjugate points, integral geometry, Crofton formula, Heber foliation, curve shortening flow

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