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Contraction de cônes complexes multidimensionnels

Abstract : The subject of this thesis is the introduction, the study and the applications of multidimensional complex cones. First, we study the grassmannian of Banach space. We define a notion of right decomposition for p-dimensional spaces and we prove the equivalence between theHausdorff distance on the grassmannian and the distance given by a norm on the exterior algebra.Then, we define p-dimensional complex cones and a gauge on the subspaces of dimension p of these cones. We show a contraction principle for thisgauge. This allows us to prove, for an operator contracting such a cone, the existence of a spectral gap which isolate the p leading eigenvaluesfrom the rest of the spectrum. We use this theory to prove a theorem of analytic regularity for Lyapunov exponents of a random product ofoperators contracting a cone. We also give a comparison between the Hausdorff distance for vector spaces and our gauge.Finally, we introduce a notion of dual cone for p-dimensional cones. In this setting, we prove that the topological properties of a cone translateinto topological properties for its dual and conversely. We complete the previous regularity theorem by proving the existence and the regularity ofa dominated splitting of the space into a "fast space" and a "slow space".
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Submitted on : Wednesday, February 20, 2019 - 3:14:06 PM
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Maxence Novel. Contraction de cônes complexes multidimensionnels. Systèmes dynamiques [math.DS]. Université Paris-Saclay, 2018. Français. ⟨NNT : 2018SACLS263⟩. ⟨tel-02042681⟩



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