Lagrangian systems on hyperbolic manifolds
Abstract
This paper gives two results that show that the dynamics of a timeperiodic Lagrangian system on a hyperbolic manifold are at least as complicated as the geodesic flow of a hyperbolic metric. Given a hyperbolic geodesic in the Poincaré ball, Theorem A asserts that there are minimizers of the lift of the Lagrangian system that are a bounded distance away and have a variety of approximate speeds. Theorem B gives the existence of a collection of compact invariant sets of the EulerLagrange flow that are semiconjugate to the geodesic flow of a hyperbolic metric. These results can be viewed as a generalization of the AubryMather theory of twist maps and the HedlundMorseGromov theory of minimal geodesics on closed surfaces and hyperbolic manifolds.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 January 1996
 arXiv:
 arXiv:math/9601212
 Bibcode:
 1996math......1212B
 Keywords:

 Mathematics  Dynamical Systems