Problem of minimal entropy for Fisler metrics

On every compact locally symmetric space of non-compact type with rank at l east two, we give an explicit construction of a Finsler metric whose total volume is the same as the Riemannian volume of the space but with a volume growth entropy strictly less than the one of the locally symmetric metric. In addition, this Finsler metric is the unique minimum for volume growth en tropy among all G-invariant Finsler metrics normalized by the volume of the manifold. On the other hand, concerning the rank one case, we prove that real hyperbo lic metrics on a compact manifold are critical points for topological entro py among all Finsler metrics normalized either by the volume of the manifol d (in all dimensions) or by the Liouville volume (on a surface).