BILLIARD REPRESENTATION FOR MULTIDIMENSIONAL COSMOLOGY WITH MULTICOMPONENT PERFECT FLUID NEAR THE SINGULARITY

The multidimensional cosmological model describing the evolution of n Einstein spaces is considered in the presence of a multicomponent perf ect fluid. When certain restrictions on the parameters of the model ar e imposed, the dynamics of the model near the singularity are 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 it s compactness is suggested. This criterion reduces the problem to the problem of illumination of an (n - 2)-dimensional sphere S-n-2 by poin t-like sources. Some generalizations of the considered scheme (includi ng scalar field and quantum generalizations) are considered.