Showing papers by "Stefano Luzzatto published in 2014"
Posted Content•
TL;DR: In this paper, the authors consider partially hyperbolic diffeomorphisms of compact Riemannian manifolds of arbitrary dimension which admit Gibbs-Markov-Young geometric structures with integrable return times.
Abstract: We consider partially hyperbolic \( C^{1+} \) diffeomorphisms of compact Riemannian manifolds of arbitrary dimension which admit a partially hyperbolic tangent bundle decomposition \( E^s\oplus E^{cu} \). Assuming the existence of a set of positive Lebesgue measure on which \( f \) satisfies a weak nonuniform expansivity assumption in the centre~unstable direction, we prove that there exists at most a finite number of transitive attractors each of which supports an SRB measure. As part of our argument, we prove that each attractor admits a Gibbs-Markov-Young geometric structure with integrable return times. We also characterize in this setting SRB measures which are liftable to Gibbs-Markov-Young structures.
28 citations
TL;DR: In this paper, the integrability of 2-dimensional invariant distributions (tangent sub-bundles) arising naturally in the context of dynamical systems on 3-manifolds was investigated.
Abstract: We investigate the integrability of 2-dimensional invariant distributions (tangent sub-bundles) which arise naturally in the context of dynamical systems on 3-manifolds. In particular we prove unique integrability of dynamically dominated and volume dominated Lipschitz continuous invariant decompositions as well as distributions with some other regularity conditions.
3 citations
Posted Content•
TL;DR: In this paper, it was shown that the direct product of maps with Young towers admits a Young tower whose return times decay at a rate bounded by the slowest of the rates of decay of the component maps.
Abstract: We show that the direct product of maps with Young towers admits a Young tower whose return times decay at a rate which is bounded above by the slowest of the rates of decay of the return times of the component maps. An application of this result, together with other results in the literature, yields various statistical properties for the direct product of various classes of systems, including Lorenz-like maps, multimodal maps, piecewise $ C^2 $ interval maps with critical points and singularities, Henon maps and partially hyperbolic systems.
1 citations
Posted Content•
TL;DR: In this paper, a notion of (uniform) asymptotic involutivity was formulated and it was shown that it implies unique integrability of corank-1 continuous distributions in dimensions three or less.
Abstract: We formulate a notion of (uniform) asymptotic involutivity and show that it implies (unique) integrability of corank-1 continuous distributions in dimensions three or less. This generalizes and extends a classical theorem of Frobenius Theorem which says that an involutive C^1 distribution is uniquely integrable.
1 citations
Posted Content•
29 Aug 2014
TL;DR: In this paper, it was shown that the direct product of maps with Young towers admits a Young tower whose return times decay at a rate bounded by the slowest of the rates of decay of the component maps.
Abstract: We show that the direct product of maps with Young towers admits a Young tower whose return times decay at a rate which is bounded above by the slowest of the rates of decay of the return times of the component maps. An application of this result, together with other results in the literature, yields various statistical properties for the direct product of various classes of systems, including Lorenz-like maps, multimodal maps, piecewise $ C^2 $ interval maps with critical points and singularities, Henon maps and partially hyperbolic systems.