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.
Cite
@article{arxiv.math/9601212,
title = {Lagrangian systems on hyperbolic manifolds},
author = {Philip Boyland and Christopher Golé},
journal= {arXiv preprint arXiv:math/9601212},
year = {2016}
}