  A quantum analysis of the vacuum Bianchi IX model is performed, focusing in
particular on the chaotic nature of the system. The framework constructed here
is general enough for the results to apply in the context of any theory of
quantum gravity, since it includes only minimal approximations that make it
possible to encode the information of all quantum degrees of freedom in the
fluctuations of the usual anisotropy parameters. These fluctuations are
described as canonical variables that extend the classical phase space. In this
way, standard methods for dynamical systems can be applied to study the chaos
of the model. Two specific methods are applied that are suitable for
time-reparameterization invariant systems. First, a generalized version of the
Misner-Chitre variables is constructed, which provides an isomorphism between
the quantum Bianchi IX dynamics and the geodesic flow on a suitable Riemannian
manifold, extending, in this way, the usual billiard picture. Secondly, the
fractal dimension of the boundary between points with different outcomes in the
space of initial data is numerically analyzed. While the quantum system remains
chaotic, the main conclusion is that its strength is considerably diminished by
quantum effects as compared to its classical counterpart.
