Abstract

Let M be a closed oriented surface endowed with a Riemannian metric g and let Ω be a 2-form. We show that the magnetic flow of the pair (g,Ω) has zero asymptotic Maslov index and zero Liouville action if and only if g has constant Gaussian curvature, Ω 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,Ω) is C¹-conjugate to the horocycle flow of a hyperbolic metric ḡ, there exists a constant a > 0 such that ag and ḡ are isometric and a⁻¹Ω 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.

Keywords

magnetic flow, horocycle flow, Aubry–Mather theory

Mathematical Subject Classification

Primary: 37D40, 53D25, 37C27

Authors