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Rigidity properties of Anosov optical hypersurfaces

Published online by Cambridge University Press:  01 June 2008

NURLAN S. DAIRBEKOV  and
GABRIEL P. PATERNAIN
NURLAN S. DAIRBEKOV
Affiliation:
Kazakh British Technical University, Tole bi 59, 050000 Almaty, Kazakhstan (email: [email protected])
GABRIEL P. PATERNAIN
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, UK (email: [email protected])
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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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 3 , June 2008 , pp. 707 - 737
Copyright
Copyright © Cambridge University Press 2008

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