Billiard representation for pseudo-Euclidean Toda-like systems of cosmological origin

Abstract

The pseudo-Euclidean Toda-like system of cosmological origin is considered. When certain restrictions on the parameters of the model are imposed, the dynamics of the model near the ``singularity'' is reduced to a billiard on the (n-1)-dimensional Lobachevsky space H^{n-1}. The geometrical criterion for the finiteness of the billiard volume and its compactness is suggested. This criterion reduces the problem to the problem of illumination of (n-2)-dimensional sphere S^{n-2} by point-like sources. Some examples are considered.