Billiards in Finsler and Minkowski geometries

We begin the study of billiard dynamics in Finsler geometry. We deduce the Finsler billiard reflection law from the "least action principle", and exte nd the basic properties of Riemannian and Euclidean billiards to the Finsle r and Minkowski settings, respectively. We prove that the Finsler billiard map is a symplectomorphism, and compute the mean free path of the Finsler b illiard ball. For the planar Minkowski billiard we obtain the min,or equati on, and extend the Mather's non-existence of caustics result. We establish an orbit-to-orbit duality for Minkowski billiards. (C) 2002 Elsevier Scienc e B.V. All rights reserved.