  The dynamics of cosmological billiards in $4=3+1$ spacetime dimensions is
analyzed; the different statistical maps are characterized within the
stochastic limit, reached after a large number of iterations of the billiard
maps. New densities of invariant measures have been established, also for
billiard systems which contain symmetry walls, according to the content of Weyl
reflections in the maps, which account for the change of sign of the
non-oscillating scale factors in the solution to the Einstein field equations.
The statistical equivalence between the big billiard and the small billiard,
posed in [Phys. Rev. D83, 044038 (2011)], is here proven by means of these new
definitions of probabilities for the small billiard. Further new classes of BKL
probabilities have also been defined especially for the one-variable map and
for the two-variable map, for the early-time BKL dynamics, for a stochastizing
BKL dynamics and for a completely stochastized dynamics, both for the big
billiard and for the small billiard. The trajectories have been classified
according to these new probabilties, and different specifications of
probabilties comparing classes of initial conditions have been assigned for the
stochastization of the dynamics. As a result, is is possible to establish a
definition of BKL probabilities for the unquotiented dynamics of the big
billiard, where the different patterns of Weyl reflections are encoded. The
statistical description of BKL probabilities for the occurrence of a given
number of epochs in each era are therefore further characterized by the most
probable number of Weyl reflections contained in such eras, which is inferred
from the implications of the billiard maps on the UPHP. These new constructions
have been considered for the determination of the connection between the
observed values of anisotropy by a stochastic limit of the BKL dynamics.
