Resumen de Magnetic rigidity of horocycle flows

Gabriel P. Paternain

• Let M be a closed oriented surface endowed with a Riemannian metric g and let O be a 2-form. We show that the magnetic flow of the pair (g,O) has zero asymptotic Maslov index and zero Liouville action if and only if g has constant Gaussian curvature, O 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,O) is C1-conjugate to the horocycle flow of a hyperbolic metric ?, there exists a constant a > 0 such that ag and ? are isometric and a-1O is, up to a sign, the area form of g. It also implies that if a magnetic flow is Mañé-critical and uniquely ergodic it must be the horocycle flow.

  As a byproduct we show the existence of closed magnetic geodesics for almost all energy levels in the case of weakly exact magnetic fields on closed manifolds of arbitrary dimension satisfying a certain technical condition