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The scaling properties of quantum gravity are discussed by employing a classof proper-time regulators in the functional flow equation for the conformalfactor within the formalism of the background field method. Renormalizationgroup trajectories obtained by projecting the flow on a flat topology are morestable than those obtained from a projection on a spherical topology. In thelatter case the ultraviolet flow can be characterized by a Hopf bifurcationwith an ultraviolet attractive limiting cycle. Although the possibility ofdetermining the infrared flow for an extended theory space can be severelyhampered due to the conformal factor instability, we present a robust numericalapproach to study the flow structure around the non-gaussian fixed point as aninverse-problem strategy. In particular it is shown the possibility of having aspontaneous breaking of the diffeomorphism invariance can be realized withnon-local functionals of the volume operator.

Universality and symmetry breaking in conformally reduced quantum gravity