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.

1 aLuzzatto, Stefano1 aTüreli, Sina1 aWar, Khadim, Mbacke uhttps://doi.org/10.1142/S0129167X1650061000676nas a2200157 4500008004100000245004500041210004500086260002100131300001000152490000700162520023500169100002200404700001800426700002400444856005000468 2016 eng d00aIntegrability of C1 invariant splittings0 aIntegrability of C1 invariant splittings bTaylor & Francis a79-880 v313 aWe 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.

1 aLuzzatto, Stefano1 aTüreli, Sina1 aWar, Khadim, Mbacke uhttps://doi.org/10.1080/14689367.2015.105748001024nas a2200121 4500008004100000245005200041210005100093260001000144520060600154653007300760100001800833856005100851 2015 en d00aIntegrability of Continuous Tangent Sub-bundles0 aIntegrability of Continuous Tangent Subbundles bSISSA3 aIn 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.10aDynamical Systems, Global Analysis, Frobenius Theorem, Integrability1 aTüreli, Sina uhttp://urania.sissa.it/xmlui/handle/1963/34630