DESCRIPTION OF STATISTICAL PROPERTIES OF THE MIXMASTER UNIVERSE

Stochastic properties of the homogeneous Bianchi type-VIII and -IX (th e mixmaster) models near the cosmological singularity are more distinc tive in the Hamiltonian formalism in the Misner-Chitre parametrization . We show how the simplest analysis of the dynamical evolution leads, in a natural way, to the construction of a stationary invariant measur e distribution which provides the complete statistical description of the stochastic behavior of these systems. We also establish the differ ence between the statistical description in the framework of the Misne r-Chitre approach and that one based on the BKL (Belinski-Khalatnikov- Lifshitz) map by means of an explicit reduction of the invariant measu re in the continuous case to the measure on the map. It turns out that the invariant measure in the continuous case contains an explicit inf ormation about durations of Kasner eras, while the measure in the case of the BKL map does not. [S0556-2821(97)00420-7].