Lipschitz Distributions and Anosov Flows

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 C1,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. The main purpose of this paper is to generalize the theorem of Frobenius on integrability of smooth vector distributions and to give an application of the theorem to the question of existence of global cross sections to Anosov flows. Accordingly, the paper is divided into two parts, A and B. A. Integrability of Lipschitz distributions Let M be a C∞ n-dimensional Riemannian manifold equipped with a Lebesgue measure. Definition 1. We will say that a distribution (or plane field) E on M is Lipschitz if it is locally spanned by Lipschitz continuous vector fields. Recall that a map f between metric spaces (M1, d1) and (M2, d2) is called Lipschitz continuous (or simply Lipschitz) if there is a constant C > 0 such that d2(f(p), f(q)) ≤ Cd1(p, q), for all p, q ∈ M1. By saying that a vector field X on M is Lipschitz we mean that in some (and therefore in any) coordinate system, X can be written in the form X = n ∑