Description of statistical properties of the mixmaster universe

Stochastic properties of the homogeneous Bianchi type-VIII and -IX (the mixmaster) models near the cosmological singularity are more distinctive in the Hamiltonian formalism in the Misner-Chitré parametrization. We show how the simplest analysis of the dynamical evolution leads, in a natural way, to the construction of a stationary invariant measure distribution which provides the complete statistical description of the stochastic behavior of these systems. We also establish the difference between the statistical description in the framework of the Misner-Chitré approach and that one based on the BKL (Belinski–Khalatnikov–Lifshitz) map by means of an explicit reduction of the invariant measure in the continuous case to the measure on the map. It turns out that the invariant measure in the continuous case contains an explicit information about durations of Kasner eras, while the measure in the case of the BKL map does not.