Hard chaos in magnetic billiards (on the hyperbolic plane)

Abstract

In this paper some results on the local and global stability analysis of magnetic billiard systems, established on two dimensional Riemannian manifolds of constant curvature are presented, with particular emphasis on the hyperbolic plane. For special billiards, possessing a discrete group of (rotational or translational) symmetry, a geometrical theorem, illustrated by numerical simulations, is given on the stability of trajectories with the same symmetry. We also present sufficient criteria for the global hyperbolicity of the dynamics (hard chaos), and give lower estimations for the Lyapunov exponent in terms of the shape of the billiard.