D.Burago, F.Ferleger, A.Kononenko
Uniform estimates on the number of collisions in semi-dispersing
billiards.
(38K, Latex 2e)
ABSTRACT. We use the methods of non-standard Riemannian geometry to obtain
complete
solutions of two classic billiard problems:
1. We establish
local uniform estimates on the number of collisions in non-degenerate
semi-dispersing billiards (moreover, our results apply to billiards
on arbitrary manifolds).
2. We find an explicit uniform estimate on the maximal possible number
of
collisions, in the infinite period of time, in an arbitrary system of
hard
spheres in empty space, in terms of the number of spheres, their masses
and
radii.
Our methods are based on the construction of a certain Alexandrov
space of
curvature bounded from above which geodesics naturally correspond to
the
trajectories of the billiard flow.