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Stable ergodicity and physical measures for weakly hyperbolic dynamical systems

Abstract : In this thesis we study the following topics:-stable ergodicity for conservative systems;-genericity of the existence of positive exponents for some skew products with two dimensional fibers;-rigidity of u-Gibbs measure for certain partially hyperbolic systems;-robust transitivity. We give a proof of stable ergodicity for a certain partially hyperbolic system without using accessibility. This system was introduced by Pierre Berger and Pablo Carrasco, and it has the following properties: it has a two dimensional center direction; it is non-uniformly hyperbolic having both a positive and a negative exponent along the center for almost every point, and the Oseledets decomposition is not dominated.In a different work, we find criteria of stable ergodicity for systems with a dominated splitting. In particular, we explore the notion of chain-hyperbolicity introduced by Sylvain Crovisier and Enrique Pujals. With this notion we give explicit criteria of stable ergodicity, and we give some applications. In a joint work with Mauricio Poletti, we prove that the random product of conservative surface diffeomorphisms generically has a region with positive exponents. Our results also hold for more general skew products. We also study dissipative perturbations of the Berger-Carrasco example. We classify all the u-Gibbs measures that may appear inside a neighborhood of the example. In this neighborhood, we prove that any u-Gibbs measure is either the unique SRB measure of the system or it has atomic disintegration along the center foliation. In a joint work with Pablo Carrasco, we prove that this example is robustly transitive (indeed robustly topologically mixing).
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Submitted on : Monday, January 27, 2020 - 6:07:44 PM
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Davi dos Anjos Obata. Stable ergodicity and physical measures for weakly hyperbolic dynamical systems. Dynamical Systems [math.DS]. Université Paris Saclay (COmUE), 2019. English. ⟨NNT : 2019SACLS488⟩. ⟨tel-02457102⟩



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