Mathematical Research Letters

Volume 12 (2005)

Number 5

Longitudinal KAM-cocycles and action spectra of magnetic flows

Pages: 719 – 730



Nurlan S. Dairbekov (Kazakh British Technical University)

Gabriel P. Paternain (University of Cambridge)


Let $M$ be a closed oriented surface and let $\Omega$ be a non-exact 2-form. Suppose that the magnetic flow $\phi$ of the pair $(g,\Omega)$ is Anosov. We show that the longitudinal KAM-cocycle of $\phi$ is a coboundary if and only if the Gaussian curvature is constant and $\Omega$ is a constant multiple of the area form thus extending the results in \cite{P2}. We also show infinitesimal rigidity of the action spectrum of $\phi$ with respect to variations of $\Omega$. Both results are obtained by showing that if $G:M\to\mathbb R$ is any smooth function and $\omega$ is any smooth $1$-form on $M$ such that $G(x)+\omega_{x}(v)$ integrates to zero along any closed orbit of $\phi$, then $G$ must be identically zero and $\omega$ must be exact.

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