  The scaling properties of quantum gravity are discussed by employing a class
of proper-time regulators in the functional flow equation for the conformal
factor within the formalism of the background field method. Renormalization
group trajectories obtained by projecting the flow on a flat topology are more
stable than those obtained from a projection on a spherical topology. In the
latter case the ultraviolet flow can be characterized by a Hopf bifurcation
with an ultraviolet attractive limiting cycle. Although the possibility of
determining the infrared flow for an extended theory space can be severely
hampered due to the conformal factor instability, we present a robust numerical
approach to study the flow structure around the non-gaussian fixed point as an
inverse-problem strategy. In particular it is shown the possibility of having a
spontaneous breaking of the diffeomorphism invariance can be realized with
non-local functionals of the volume operator.
