Hölder Forms and Integrability of Invariant Distributions

Abstract. We prove an inequality for Hölder continuous differential forms on compact manifolds in which the integral of the form over the boundary of a sufficiently small, smoothly immersed disk is bounded by a certain multiplicative convex combination of the volume of the disk and the area of its boundary. This inequality has natural applications in dynamical systems, where Hölder forms are ubiquitous. We give four such applications. In the first two we prove criteria for the existence of global cross sections to Anosov flows in terms of their expansioncontraction rates and Lyapunov exponents. The second two applications provide analogous criteria for non-accessibility of partially hyperbolic diffeomorphisms.