Longitudinal KAM-cocycles and action spectra of magnetic flows

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 the Gaussian curvature is constant and $Ω$ 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 $ϕ$ with respect to variations of $Ω$. Both results are obtained by showing that if $G:M\to\mathbb R$ 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.