1,294 research outputs found

    Partially hyperbolic dynamics in dimension 3

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    Partial hyperbolicity appeared in the sixties as a natural generaliza- tion of hyperbolicity. In the last 20 years in this area there has been great activity. Here we survey the state of the art in some topics, focusing especially in partial hyperbolicity in dimension 3. The reason for this is not only that it is the smallest dimension in which non-degenerate partial hyperbolicity can occur, but also that the topology of 3-manifolds influences this dynamics in revealing ways.Comment: Updated version, to appear in Ergodic Theory and Dynamical System

    On m-minimal partially hyperbolic diffeomorphisms

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    We discuss about the denseness of the strong stable and unstable manifolds of partially hyperbolic diffeomorphisms. In this sense, we introduce a concept of m-minimality. More precisely, we say that a partially hyperbolic diffeomorphisms is m-minimal if m-almost every point in M has its strong stable and unstable manifolds dense in M. We show that this property has dynamics consequences: topological and ergodic. Also, we prove the abundance of m-minimal partially hyperbolic diffeomorphisms in the volume preserving and symplectic scenario

    A survey on partially hyperbolic dynamics

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    Some of the guiding problems in partially hyperbolic systems are the following: (1) Examples, (2) Properties of invariant foliations, (3) Accessibility, (4) Ergodicity, (5) Lyapunov exponents, (6) Integrability of central foliations, (7) Transitivity and (8) Classification. Here we will survey the state of the art on these subjects, and propose related problems.Comment: 57 pages, references adde

    Ergodicity of partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds

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    We study conservative partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds. We show that they are always accessible and deduce as a result that every conservative C1+C^{1+} partially hyperbolic in a hyperbolic 3-manifold must be ergodic, giving an afirmative answer to a conjecture of Hertz-Hertz-Ures in the context of hyperbolic 3-manifolds. Some of the intermediary steps are also done for general partially hyperbolic diffeomorphisms homotopic to the identity.Comment: 40 pages, 4 figures. New version takes into account the release of arXiv:1908.06227 while the previous version made reference to an earlier (and unavailable) version of that paper. Also adds new results on certain isotopy classes on Seifert manifold

    Some advances on generic properties of the Oseledets splitting

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    In his foundational paper [ICM 1983, Warzaw], Ma\~n\'e suggested that some aspects of the Oseledets splitting could be improved if one worked under C1-generic conditions. He announced some powerful theorems, and suggested some lines to follow. Here we survey the state of the art and some recent advances in these directions.Comment: 22 page

    Robust transitivity implies almost robust ergodicity

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    In this paper we show the relation between robust transitivity and robust ergodicity for conservative diffeomorphisms. In dimension 2 robustly transitive systems are robustly ergodic. For the three dimensional case, we define it almost robust ergodicity and prove that generically robustly transitive systems are almost robustly ergodic, if the Lyapunov exponents are nonzero. We also show in higher dimensions, that under some conditions robust transitivity implies robust ergodicity.Comment: 12 page

    A Criterion For Ergodicity of Non-uniformly hyperbolic Diffeomorphisms

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    In this work we exhibit a new criteria for ergodicity of diffeomorphisms involving conditions on Lyapunov exponents and general position of some invariant manifolds. On one hand we derive uniqueness of SRB-measures for transitive surface diffeomorphisms. On the other hand, using recent results on the existence of blenders we give a positive answer, in the C1C^1 topology, to a conjecture of Pugh-Shub in the context of partially hyperbolic conservative diffeomorphisms with two dimensional center bundle.Comment: A Research Announcemen

    Partial hyperbolicity and ergodicity in dimension three

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    In [15] the authors proved the Pugh-Shub conjecture for partially hyperbolic diffeomorphisms with 1-dimensional center, i.e. stable ergodic diffeomorphism are dense among the partially hyperbolic ones. In this work we address the issue of giving a more accurate description of this abundance of ergodicity. In particular, we give the first examples of manifolds in which all conservative partially hyperbolic diffeomorphisms are ergodic.Comment: 14 page

    On The Uniqueness of SRB Measures for Endomorphisms

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    In this paper we improve the results of \cite{MT} and show that a weak hyperbolic transitivity implies the uniqueness of hyperbolic SRB measures. As an important corollary, it arises the ergodicity of the system in a conservative setting. It also arises the condition which implies the stable ergodicity as well as the statistical stability for a general C2C^2-regular map

    On the stable ergodicity of diffeomorphisms with dominated splitting

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    In this paper we obtain two criteria of stable ergodicity outside the partially hyperbolic scenario. In both criteria, we use a weak form of hyperbolicity called chain-hyperbolicity. It is obtained one criterion for diffeomorphisms with dominated splitting and one criterion for weakly partially hyperbolic diffeomorphisms. As an application of one of these criteria, we obtain the C1C^1-density of stable ergodicity inside a certain class of weakly partially hyperbolic diffeomorphisms
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