  We analyze the Bianchi IX Universe in the Polymer Quantum Mechanics framework
by facing both semiclassical and purely quantum effects near the cosmological
singularity. We adopt Misner variables to describe the model dynamics, applying
the polymer paradigm simultaneously to the isotropic one (linked to the
Universe volume) and to the two anisotropy ones (characterizing the physical
gravitational degrees of freedom). Setting two different cut-off scales for the
two different variable sets, i.e. the geometrical volume and the gravity tensor
modes, we demonstrate how the semiclassical properties of the Bianchi IX
dynamics are sensitive to the ratio of the cut-off parameters. In particular,
the semiclassical evolution turns out to be chaotic only if the parameter
associated to the volume discretization is greater or equal to that one of the
anisotropies. Concerning the chaotic case we perform a purely quantum polymer
analysis, demonstrating that the original Misner result about the existence of
quasi-classical states near the singularity (in the sense of high occupation
numbers) is still valid in the revised approach and able to account for cut-off
physics effects. The possibility for a comparison with the original study by
Misner is possible because the singularity is still present in the
semiclassical evolution of the cosmological model for all the parameter space.
We interpret this surprising feature as the consequence of a geometrical volume
discretization which does not prevent the volume from vanishing, i.e. restoring
in the Minisuperspace analysis its zero value.
