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

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