On Finsler manifolds of negative flag curvature Yong Fang and Patrick Foulon D´epartement de Math´ematiques,Universit´ede Cergy-Pontoise, 2, avenue Adolphe Chauvin 95302 Cergy-Pontoise Cedex, France (e-mail :
[email protected]) CIRM, Luminy, case 916, 13288 Marseille Cedex 9, France (e-mail:
[email protected]) Abstract One of the key differences between Finsler metrics and Rieman- nian metrics is the non-reversibility, i.e. given two points p and q, the Finsler distance d(p; q) is not necessarily equal to d(q; p). In this pa- per, we build the main tools to investigate the non-reversibility in the context of large-scale geometry of uniform Finsler Cartan-Hadamard manifolds. In the second part of this paper, we use the large-scale geometry to prove the following dynamical theorem: Let ' be the geodesic flow of a closed negatively curved Finsler manifold. If its Anosov splitting is C2, then its cohomological pressure is equal to its Liouville metric entropy. This result generalizes a previous Riemannian result of U. Hamenst¨adt. 1 Introduction Let M be a C1 manifold and TM be its tangent bundle. A C1-Finsler + metric on M is a function F : TM ! R satisfying: (a) F (tu) = tF (u) for any u 2 TM and t ≥ 0; (b) F is strictly positive and C1 over TM − f0g; (c) In standard local coordinates (xi; yi) on TM, the matrix of partial deriva- 2 2 tives @ F is positive definite. @yi@yj A Finsler metric F is said to be Riemannian if there exists a C1- p Riemannian metric g such that F = g.