  We analyze the quantum dynamics of the Bianchi Type IX model, as described in
the so-called polymer representation of quantum mechanics, to characterize the
modifications that a discrete na- ture in the anisotropy variables of the
Universe induces on the morphology of the cosmological sin- gularity. We first
perform a semiclassical analysis, to be regarded as the zeroth-order
approximation of a WKB (Wentzel-Kramers-Brillouin) approximation of the quantum
dynamics, and demonstrate how the features of polymer quantum mechanics are
able to remove the chaotic properties of the Bianchi IX dynamics. Then, we
address the study of the full quantum dynamics of this model in the polymer
representation and analyze the two cases, in which the Bianchi IX spatial
curvature does not affect the wave-packet behavior, as well as the instance,
for which it plays the role of an infinite potential confining the dynamics of
the anisotropic variables. The main development of this analysis consists of
investigating how, differently from the standard canonical quantum evolution,
the high quantum number states are not preserved arbitrarily close to the
cosmological singularity. This property emerges as a consequence, on one hand,
of the no longer chaotic features of the classical dynamics (on which the
Misner analysis is grounded), and, on the other hand, of the impossibility to
remove the quantum effect due to the spatial curvature. In the polymer picture,
the quantum evolution of the Bianchi IX model remains always significantly far
from the semiclassical behavior, as far as both the wave-packet spread and the
occupation quantum numbers are concerned. As a result, from a quantum point of
view, the Mixmaster dynamics loses any predictivity characterization for the
discrete nature of the Universe anisotropy.
