We investigate the integrability of two-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.

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 Frobenius theorem, which says that an involutive C1 distribution is uniquely integrable.

%B International Journal of Mathematics %V 27 %P 1650061 %G eng %U https://doi.org/10.1142/S0129167X16500610 %R 10.1142/S0129167X16500610 %0 Journal Article %J Dynamical Systems %D 2016 %T Integrability of C1 invariant splittings %A Stefano Luzzatto %A Sina Türeli %A Khadim Mbacke War %XWe derive some new conditions for integrability of dynamically defined C1 invariant splittings, formulated in terms of the singular values of the iterates of the derivative of the diffeomorphism which defines the splitting.

%B Dynamical Systems %I Taylor & Francis %V 31 %P 79-88 %G eng %U https://doi.org/10.1080/14689367.2015.1057480 %R 10.1080/14689367.2015.1057480 %0 Journal Article %J Discrete & Continuous Dynamical Systems - A %D 2016 %T Young towers for product systems %A Stefano Luzzatto %A Marks Ruziboev %XWe 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, Hénon maps and partially hyperbolic systems.

%B Discrete & Continuous Dynamical Systems - A %V 36 %P 1465 %G eng %U http://aimsciences.org//article/id/18d4526e-470d-467e-967a-a0345ad4c642 %R 10.3934/dcds.2016.36.1465