Magnetic rigidity of horocycle flows

Gabriel P. Paternain

Vol. 225 (2006), No. 2, 301–323

DOI: 10.2140/pjm.2006.225.301

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 $\left(g,\Omega \right)$ has zero asymptotic Maslov index and zero Liouville action if and only if $g$ 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 $\left(g,\Omega \right)$ is $C^1$-conjugate to the horocycle flow of a hyperbolic metric $ḡ$, there exists a constant $a>0$ such that $ag$ and $ḡ$ are isometric and $a^{-1}\Omega$ 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 2000

Primary: 37D40, 53D25, 37C27

Milestones

Received: 8 October 2004

Revised: 23 May 2005

Accepted: 9 June 2005

Published: 1 June 2006