## Institute for Mathematical Sciences

## Preprint ims96-1a

** P. Boyland and C. Gole**
* Lagrangian Systems on Hyperbolic Manifolds*

Abstract: 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 the
Poincar\'e 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 Euler-Lagrange flow that are
semiconjugate to the geodesic flow of a hyperbolic metric.
These results can be viewed as a generalization of the
Aubry-Mather theory of twist maps and the
Hedlund-Morse-Gromov theory of minimal geodesics on closed
surfaces and hyperbolic manifolds.

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