  According to the Belinski-Khalatnikov-Lifshitz conjecture, close to a
spacelike singularity different spatial points decouple, and the dynamics can
be described in terms of the Mixmaster (vacuum Bianchi IX) model. In order to
understand the role played by quantum-gravity effects in this context, in the
present work we consider the semiclassical behavior of this model. Classically,
this system undergoes a series of transitions between Kasner epochs, which are
described by a specific transition law. This law is derived based on the
conservation of certain physical quantities and the rotational symmetry of the
system. In a quantum scenario, however, fluctuations and higher-order moments
modify these quantities, and consequently also the transition rule. In
particular, we perform a canonical quantization of the model and then
analytically obtain the modifications of this transition law for semiclassical
states peaked around classical trajectories. The transition rules for the
quantum moments are also obtained, and a number of interesting properties are
derived concerning the coupling between the different degrees of freedom. More
importantly, we show that, due to the presence of quantum-gravity effects and
contrary to the classical model, there appear certain finite ranges of the
Kasner parameters for which the system will not undergo further transitions and
will follow a Kasner regime until the singularity. Even if a more detailed
analysis is still needed, this feature points toward a possible resolution of
the classical chaotic behavior of the model. Finally, a numerical integration
of the equations of motion is also performed in order to verify the obtained
analytical results.
