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Article · Ergodic Theory and Dynamical Systems · March 19, 2008

Rigidity properties of Anosov optical hypersurfaces

N. S. Dairbekov, Gabriel P. Paternain

Nurlan Dairbekov
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Abstract

Abstract We consider an optical hypersurface Σ in the cotangent bundle τ:T*M→M of a closed manifold M endowed with a twisted symplectic structure. We show that if the characteristic foliation of Σ is Anosov, then a smooth 1-form θ on M is exact if and only if τ*θ has zero integral over every closed characteristic of Σ. This result is derived from a related theorem about magnetic flows which generalizes our previous work [N. S. Dairbekov and G. P. Paternain. Longitudinal KAM cocycles and action spectra of magnetic flows. Math. Res. Lett.12 (2005), 719–729]. Other rigidity issues are also discussed.

Mathematical Dynamics and FractalsGeometric and Algebraic TopologyAdvanced Differential Geometry Researchcotangent bundlecharacteristic foliationrigidity

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