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Theses

Quantitative recurrence properties in infinite measure

Abstract : In this thesis, we study the quantitative recurrence properties of some dynamical systems preserving an infinite measure. We are interested in the first return time of the orbits of a dynamical system into a small neighborhood of their starting points. First, we start by considering a toy probabilistic model to clarify the strategy of our proofs. Our interest is when the measure is indeed infinite, more precisely we consider the Z-extensions of subshifts of finite type. We study the asymptotic behavior of the first return time near the origin, and we establish results of an almost everywhere convergence kind, and a convergence in distribution with respect to any probability measure absolutely continuous with respect to the infinite measure. In this work, we are also interested in another dynamicals systems. We consider an Axiom A flow (gt)t on a Riemannian manifold M endowed with a σ-finite measure μ. We will assume that the measure μ is an equilibrium measure for (gt)t. In order to establish our results, we introduce notions from hyperbolic dynamics. In particular, we consider the Markov section which was constructed by Bowen and Ratner.
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Nasab Yassine. Quantitative recurrence properties in infinite measure. Dynamical Systems [math.DS]. Université de Bretagne occidentale - Brest, 2018. English. ⟨NNT : 2018BRES0062⟩. ⟨tel-02081262⟩

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