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.

VL - 27 UR - https://doi.org/10.1142/S0129167X16500610 ER - TY - JOUR T1 - Integrability of C1 invariant splittings JF - Dynamical Systems Y1 - 2016 A1 - Stefano Luzzatto A1 - Sina Türeli A1 - Khadim Mbacke War AB -We 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.

PB - Taylor & Francis VL - 31 UR - https://doi.org/10.1080/14689367.2015.1057480 ER - TY - THES T1 - Integrability of Continuous Tangent Sub-bundles Y1 - 2015 A1 - Sina Türeli KW - Dynamical Systems, Global Analysis, Frobenius Theorem, Integrability AB - In this thesis, the main aim is to study the integrability properties of continuous tangent sub-bundles, especially those that arise in the study of dynamical systems. After the introduction and examples part we start by studying integrability of such sub-bundles under different regularity and dynamical assumptions. Then we formulate a continuous version of the classical Frobenius theorem and state some applications to such bundles, to ODE and PDE. Finally we close of by stating some ongoing work related to interactions between integrability, sub-Riemannian geometry and contact geometry. PB - SISSA UR - http://urania.sissa.it/xmlui/handle/1963/34630 U1 - 34833 U2 - Mathematics U5 - MAT/05 ER -