Longitudinal KAM-cocycles and action spectra of magnetic flows

N. S. Dairbekov, Gabriel P. Paternain

2005

DOI 10.4310/mrl.2005.v12.n5.a9

Data confidence 70 medium

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Abstract

Let M be a closed oriented surface and let Ω be a non-exact 2-form.Suppose that the magnetic flow φ of the pair (g, Ω) is Anosov.We show that the longitudinal KAM-cocycle of φ is a coboundary if and only if the Gaussian curvature is constant and Ω is a constant multiple of the area form thus extending the results in [12].We also show infinitesimal rigidity of the action spectrum of φ with respect to variations of Ω.Both results are obtained by showing that if G : M → R is any smooth function and ω is any smooth 1-form on M such that G(x) + ω x (v) integrates to zero along any closed orbit of φ, then G must be identically zero and ω must be exact.