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Theses

Étude de systèmes dynamiques avec perte de régularité

Abstract : The aim of this thesis is the development of a unified framework to study the regularity of certain characteristics elements of chaotic dynamics (Topological presure/entropy, Gibbs measure, Lyapunov exponents) with respect to the dynamic itself. The main technical issue is the regularity loss occuring from the use of a composition operator, the transfer operator, whose spectral properties are intimately connected to the aformentionned "characteristics elements". To overcome this issue, we developped a regularity theorem for fixed points (with respect to parameter), in the spirit of the implicit function theorem of Nash and Moser. We then apply this "fixed point" approach to the linear response problem (studying the regularity of the system invariant measure w.r.t parameters) for a family of uniformly expanding maps. In a second time, we study the regularity of the top characteristic exponent of a random prduct of expanding maps, building from our regularity theorem and cone contraction theory. We deduce from this regularity w.r.t parameters for the stationanry measure, the variance in the central limit theorem, and other quantities of dynamical interest.
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Julien Sedro. Étude de systèmes dynamiques avec perte de régularité. Systèmes dynamiques [math.DS]. Université Paris-Saclay, 2018. Français. ⟨NNT : 2018SACLS254⟩. ⟨tel-01957559⟩

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