Lagrangian systems on hyperbolic manifolds

This paper gives two results that show that the dynamics of a time-periodic Lagrangian system on a hyperbolic manifold are at least as complicated as the geodesic flow of a hyperbolic metric. Given a hyperbolic geodesic in th e Poincare 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 com pact invariant sets of the Euler-Lagrange flow that are semiconjugate to th e geodesic flow of a hyperbolic metric.: These results can be viewed as a g eneralization of the Aubry-Mather theory of twist maps and the Hedlund-Mors e-Gromov theory of minimal geodesics on closed surfaces and hyperbolic mani folds.