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Expansive geodesic flows on compact manifolds without conjugate points.

Abstract : This thesis is divided in two independants parts.In the first part we investigate dynamical properties of expansive geodesic flows on compact manifolds without conjugate points using the work of R.O.~Ruggiero. More precisely we show that such a flow admits a unique measure of maximal entropy and constructthis measure. This extends results known in non-positively curved manifolds of rank one (and our construction is analogous). Wethen show, using this measure of maximal entropy, that the asymptotics of Margulis (known for compact negatively curvedmanifolds) on the number of geodesic loops still hold in this framework.In the second part we study isometries of finite dimensionalsymmetric cones for both the Thompson and the Hilbert metric. More precisely we show that the isometry group induced by the linear automorphisms preserving such a cone is a subgroup of finite indexin the full group of isometries for those two metrics and give a natural set of representatives of the quotient. This extends resultsof L.~Molnar (who studied such isometies for the symmetric irreducible cone of symmetric positive definite operators on acomplex Hilbert space).
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Aurélien Bosché. Expansive geodesic flows on compact manifolds without conjugate points.. Complex Variables [math.CV]. Ruhr-Universität, 2015. English. ⟨NNT : 2015GREAM085⟩. ⟨tel-01691107⟩



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