Magnetic rigidity of horocycle flows

• Autores: Gabriel P. Paternain
• Localización: Pacific journal of mathematics, ISSN 0030-8730, Vol. 225, Nº 2, 2006, págs. 301-324
• Idioma: inglés
• DOI: 10.2140/pjm.2006.225.301
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• Resumen

  • 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