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In the first part of this dissertation, we give a new definition of a Laplaceoperator for Finsler metric as an average, with regard to an angle measure, ofthe second directional derivatives. This operator is elliptic, symmetric withrespect to the Holmes-Thompson volume, and coincides with the usualLaplace--Beltrami operator when the Finsler metric is Riemannian. We computeexplicit spectral data for some Katok-Ziller metrics. When the Finsler metricis negatively curved, we show, thanks to a result of Ancona that the Martinboundary is H\"older-homeomorphic to the visual boundary. This allow us todeduce the existence of harmonic measures and some ergodic preoperties. In thesecond part of this dissertation, we study Anosov flows in 3-manifolds, withleaf-spaces homeomorphic to $\mathbb{R}$. When the manifold is hyperbolic,Thurston showed that the (un)stable foliations induces a regulating flow. Weuse this second flow to study isotopy class of periodic orbits of the Anosovflow and existence of embedded cylinders.

A new Laplace operator in Finsler geometry and periodic orbits of Anosov flows