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

We show that the space of (reversible) Finsler metrics on the two-torus <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>𝕋</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>𝕋</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> without conjugate points, whose Heber foliations are smooth and with the same marked length spectrum, have conjugate geodesic flows.