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July  2009, 25(2): 669-685. doi: 10.3934/dcds.2009.25.669

Hölder forms and integrability of invariant distributions

Slobodan N. Simić 1,

1. 

Department of Mathematics, San José State University, San José, CA 95192-0103, United States

Received  June 2008 Revised  January 2009 Published  June 2009

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 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.
Keywords: global cross section, invariant distribution, Hölder forms.
Mathematics Subject Classification: Primary: 37D20, 37D30; Secondary: 49Q1.
Citation: Slobodan N. Simić. Hölder forms and integrability of invariant distributions. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 669-685. doi: 10.3934/dcds.2009.25.669
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