Lipschitz distributions and Anosov flows

Abstract

We show that if a distribution is locally spanned by Lipschitz vector fields and is involutive a.e., then it is uniquely integrable giving rise to a Lipschitz foliation with leaves of class $C^{1, \text {Lip}}$. As a consequence, we show that every codimension-one Anosov flow on a compact manifold of dimension $>3$ such that the sum of its strong distributions is Lipschitz, admits a global cross section.