  Belinski, Khalatnikov and Lifshitz (BKL) pioneered the study of the
statistical properties of the never-ending oscillatory behavior (among
successive Kasner epochs) of the geometry near a space-like singularity. We
show how the use of a "cosmological billiard" description allows one to refine
and deepen the understanding of these statistical properties. Contrary to
previous treatments, we do not quotient the dynamics by its discrete symmetry
group (of order 6), thereby uncovering new phenomena, such as correlations
between the successive billiard corners in which the oscillations take place.
Starting from the general integral invariants of Hamiltonian systems, we show
how to construct invariant measures for various projections of the
cosmological-billiard dynamics. In particular, we exhibit, for the first time,
a (non-normalizable) invariant measure on the "Kasner circle" which
parametrizes the exponents of successive Kasner epochs. Finally, we discuss the
relation between: (i) the unquotiented dynamics of the Bianchi IX (a, b, c or
mixmaster) model; (ii) its quotienting by the group of permutations of (a, b,
c); and (iii) the billiard dynamics that arose in recent studies suggesting the
hidden presence of Kac-Moody symmetries in cosmological billiards.
