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Lagrangian systems on hyperbolic manifolds

Published online by Cambridge University Press:  01 October 1999

PHILIP BOYLAND  and
CHRISTOPHE GOLÉ
PHILIP BOYLAND
Affiliation:
Department of Mathematics, University of Florida, PO Box 118105, Gainesville, FL 32611-8105, USA (e-mail: [email protected])
CHRISTOPHE GOLÉ
Affiliation:
Mathematics Department, Smith College, Northampton, MA 01063, USA (e-mail: [email protected])
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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 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 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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 19 , Issue 5 , October 1999 , pp. 1157 - 1173
Copyright
1999 Cambridge University Press

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