Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows

Yong Fang; Patrick Foulon; Boris Hasselblatt

doi:10.3934/era.2010.17.80

Article Contents

2010, Volume 17: 80-89. Doi: 10.3934/era.2010.17.80

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Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows

1.
Département de Mathématiques, Université de Cergy-Pontoise, avenue Adolphe Chauvin, 95302, Cergy-Pontoise Cedex
2.
Institut de Recherche Mathematique Avancée, UMR 7501 du Centre National de la Recherche Scientifique, 7 Rue René Descartes, 67084, Strasbourg Cedex
3.
Department of Mathematics, Tufts University, Medford, MA 02155

Received: May 2010

Published: January 2010

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Abstract

In several contexts the defining invariant structures of a hyperbolic dynamical system are smooth only in systems of algebraic origin, and we prove new results of this smooth rigidity type for a class of flows.
For a transversely symplectic uniformly quasiconformal $C^2$ Anosov flow on a compact Riemannian manifold we define the longitudinal KAM-cocycle and use it to prove a rigidity result: The joint stable/unstable subbundle is Zygmund-regular, and higher regularity implies vanishing of the KAM-cocycle, which in turn implies that the subbundle is Lipschitz-continuous and indeed that the flow is smoothly conjugate to an algebraic one. To establish the latter, we prove results for algebraic Anosov systems that imply smoothness and a special structure for any Lipschitz-continuous invariant 1-form.
We obtain a pertinent geometric rigidity result: Uniformly quasiconformal magnetic flows are geodesic flows of hyperbolic metrics.
Several features of the reasoning are interesting: The use of exterior calculus for Lipschitz-continuous forms, that the arguments for geodesic flows and infranilmanifoldautomorphisms are quite different, and the need for mixing as opposed to ergodicity in the latter case.

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Mathematics Subject Classification: Primary: 37D20, 37D40; Secondary: 53C24, 53D25.

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References

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