Integrability of Continuous Bundles

TL;DR

This paper establishes new conditions for the integrability and uniqueness of continuous tangent sub-bundles on manifolds, extending classical theorems and applying to dynamical systems and differential equations.

Contribution

It generalizes Frobenius' Theorem to less regular bundles and provides new criteria for solution uniqueness and invariant bundle integrability.

Findings

New sufficient conditions for integrability of continuous bundles

A novel proof of the Stable Manifold Theorem

Integrability results for dominated splittings in dynamics

Abstract

We give new sufficient conditions for the integrability and unique integrability of continuous tangent sub-bundles on manifolds of arbitrary dimension, generalizing Frobenius' classical Theorem for C^1 sub-bundles. Using these conditions we derive new criteria for uniqueness of solutions to ODE's and PDE's and for the integrability of invariant bundles in dynamical systems. In particular we give a novel proof of the Stable Manifold Theorem and prove some integrability results for dynamically defined dominated splittings.