  Cosmological billiards arise as a map of the solution to the Einstein
equations, when the most general symmetry of the metric tensor is implemented,
under the BKL (named after Belinskii, Khalatnikov and Lifshitz) paradigm, for
which points are spatially decoupled in the asymptotical limit close to the
cosmological singularity. Cosmological billiards in $4=3+1$ dimensions for the
case of pure gravity are analyzed for those features, for which the content of
Weyl reflections in the BKL maps requires definition of a 3-dimensional
restricted phase space. The role of Poincar\'e sections in these processes is
outlined. The quantum regime is investigated within this framework: as a
result, 1-epoch BKL eras are found to be the most probable configuration at
which the wavefunctions have to be evaluated; furthermore, BKL eras containing
$n>>1$ epochs are shown to be a less probable configuration for the
wavefunctions. This description of the dynamics allows one to gain information
about the connections between the statistical characterization of the maps
which imply the different symmetry-quotienting mechanisms and the
characterization of the semiclassical limit of the wavefunctions for the
classical trajectories, for which the phenomenon of 'scars' on the wavefunction
is found for other kinds of billiards.
