Dynamique d'action de groupes dans des espaces homogènes de rang supérieur et de volume infini

Abstract : Let G be a semisimple Lie group (of higher rank) and Γ a Zariski dense subgroup of G (of infinite covolume). In this thesis, we discuss two questions related to the Benoist limit cone of Γ: one concerns random walks, the other topological mixing of the directional Weyl chamber flow. In the introduction, we state the main results of this thesis in their context. In the second chapter, we recall some general facts about Lie groups and loxodromic elements. In the third chapter, we prove that every point of the interior of the limit cone is a Lyapunov vector. In the fourth chapter, we construct local coordinates of G and give key tools for the remaining parts. In the fifth chapter, we introduce the invariant subsets of G. In the last chapter of this thesis, we prove the topological mixing criterion of regular directional Weyl chamber flow obtained with O. Glorieux and we generalize this criterion to Γ\G for a class of Lie groups including SL(n,R), SL(n,C), SO(p,p+2)⁰.
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Nguyen-Thi Dang. Dynamique d'action de groupes dans des espaces homogènes de rang supérieur et de volume infini. Systèmes dynamiques [math.DS]. Université de Rennes 1, 2019. Français. ⟨tel-02301728⟩

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