Lagrangian Systems on Hyperbolic Manifolds

Authors

Philip Boyland, University of Florida
Christophe Golé, Smith CollegeFollow

Document Type

Article

Publication Date

1-1-1999

Publication Title

Ergodic Theory and Dynamical Systems

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 geodesies on closed surfaces and hyperbolic manifolds.

Volume

19

Issue

5

First Page

1157

Last Page

1173

DOI

10.1017/S0143385799133893

ISSN

01433857

Comments

Peer reviewed accepted manuscript.

Recommended Citation

Boyland, Philip and Golé, Christophe, "Lagrangian Systems on Hyperbolic Manifolds" (1999). Mathematics Sciences: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/81