Magnetic Rigidity of Horocycle flows

Abstract

Let $M$ be a closed oriented surface endowed with a Riemannian metric $g$ and let $\Omega$ be a 2-form. We show that the magnetic flow of the pair $(g,\Omega)$ has zero asymptotic Maslov index and zero Liouville action if and only $g$ has constant Gaussian curvature, $\Omega$ is a constant multiple of the area form of $g$ and the magnetic flow is a horocycle flow. This characterization of horocycle flows implies that if the magnetic flow of a pair $(g,\Omega)$ is $C^1$-conjugate to the horocycle flow of a hyperbolic metric $\bar{g}$ then there exists a constant $a>0$, such that $ag$ and $\bar{g}$ are isometric and $a^{-1}\Omega$ is, up to a sign, the area form of $g$. The characterization also implies that if a magnetic flow is Ma\~n\'e critical and uniquely ergodic it must be the horocycle flow. As a by-product we also obtain results on existence of closed magnetic geodesics for almost all energy levels in the case weakly exact magnetic fields on arbitrary manifolds.