H\"older forms and integrability of invariant distributions

TL;DR

This paper establishes an inequality for H"older continuous differential forms on compact manifolds, with applications to dynamical systems such as criteria for global cross sections in Anosov flows and non-accessibility in partially hyperbolic diffeomorphisms.

Contribution

It introduces a new inequality for H"older forms and applies it to derive criteria for key properties in dynamical systems.

Findings

Inequality bounds integrals of H"older forms by volume and boundary area.

Criteria for existence of global cross sections in Anosov flows.

Criteria for non-accessibility in partially hyperbolic systems.

Abstract

We prove an inequality for H\"older 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 H\"older continuity is ubiquitous. We give two such applications. In the first one, we prove a criterion for the existence of global cross sections to Anosov flows in terms of their expansion-contraction rates. The second application provides an analogous criterion for non-accessibility of partially hyperbolic diffeomorphisms.