Journal of Modern Dynamics (JMD)

Lipschitz continuous invariant forms for algebraic Anosov systems

Pages: 571 - 584, Issue 3, July 2010      doi:10.3934/jmd.2010.4.571

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Patrick Foulon - Institut de Recherche Mathematique Avancée, UMR 7501 du Centre National de la Recherche Scientifique, 7 Rue René Descartes, 67084, Strasbourg Cedex, France (email)
Boris Hasselblatt - Department of Mathematics, Tufts University, Medford, MA 02155, United States (email)

Abstract: We prove results for algebraic Anosov systems that imply smoothness and a special structure for any Lipschitz continuous invariant $1$-form. This has corollaries for rigidity of time-changes, and we give a particular application to geometric rigidity of quasiconformal Anosov flows.
   Several features of the reasoning are interesting; namely, the use of exterior calculus for Lipschitz continuous forms, the arguments for geodesic flows and infranilmanifoldautomorphisms are quite different, and the need for mixing as opposed to ergodicity in the latter case.

Keywords:  Anosov flow, invariant forms, Lipschitz regularity, smooth rigidity.
Mathematics Subject Classification:  Primary: 37D20, 37D40; Secondary: 53C24, 53D25.

Received: May 2010;      Revised: September 2010;      Available Online: October 2010.