Journal of Modern Dynamics (JMD)

Zygmund strong foliations in higher dimension

Pages: 549 - 569, Issue 3, July 2010      doi:10.3934/jmd.2010.4.549

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Yong Fang - Département de Mathématiques, Université de Cergy-Pontoise, avenue Adolphe Chauvin, 95302, Cergy-Pontoise Cedex, France (email)
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: For a compact Riemannian manifold $M$, $k\ge2$ and a uniformly quasiconformal transversely symplectic $C^k$ Anosov flow $\varphi$:$\R\times M\to M$ we define the longitudinal KAM-cocycle and use it to prove a rigidity result: $E^u\oplus E^s$ is Zygmund-regular, and higher regularity implies vanishing of the longitudinal KAM-cocycle, which in turn implies that $E^u\oplus E^s$ is Lipschitz-continuous. Results proved elsewhere then imply that the flow is smoothly conjugate to an algebraic one.

Keywords:  Anosov flow, strong invariant subbundles, quasiconformal, Zygmund regularity, smooth rigidity, geometric rigidity, foliations.
Mathematics Subject Classification:  Primary: 37D20, 37D40; Secondary: 53C24, 53D25.

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