29,359 research outputs found

    Gibbs-Markov structures and limit laws for partially hyperbolic attractors with mostly expanding central direction

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    We consider a partially hyperbolic set KK on a Riemannian manifold MM whose tangent space splits as TKM=EcuβŠ•EsT_K M=E^{cu}\oplus E^{s}, for which the centre-unstable direction EcuE^{cu} expands non-uniformly on some local unstable disk. We show that under these assumptions ff induces a Gibbs-Markov structure. Moreover, the decay of the return time function can be controlled in terms of the time typical points need to achieve some uniform expanding behavior in the centre-unstable direction. As an application of the main result we obtain certain rates for decay of correlations, large deviations, an almost sure invariance principle and the validity of the Central Limit Theorem.Comment: 23 page

    Random perturbations of non-uniformly expanding maps

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    We give both sufficient conditions and necessary conditions for the stochastic stability of non-uniformly expanding maps either with or without critical sets. We also show that the number of probability measures describing the statistical asymptotic behaviour of random orbits is bounded by the number of SRB measures if the noise level is small enough. As an application of these results we prove the stochastic stability of certain classes of non-uniformly expanding maps introduced in \cite{V} and \cite{ABV}.Comment: 44 pages, 2 figure

    Strong stochastic stability for non-uniformly expanding maps

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    We consider random perturbations of discrete-time dynamical systems. We give sufficient conditions for the stochastic stability of certain classes of maps, in a strong sense. This improves the main result in J. F. Alves, V. Araujo, Random perturbations of non-uniformly expanding maps, Asterisque 286 (2003), 25--62, where it was proved the convergence of the stationary measures of the random process to the SRB measure of the initial system in the weak* topology. Here, under slightly weaker assumptions on the random perturbations, we obtain a stronger version of stochastic stability: convergence of the densities of the stationary measures to the density of the SRB measure of the unperturbed system in the L1-norm. As an application of our results we obtain strong stochastic stability for two classes of non-uniformly expanding maps. The first one is an open class of local diffeomorphisms introduced in J. F. Alves, C. Bonatti, M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math. 140 (2000), 351--398, and the second one the class of Viana maps.Comment: 43 page
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