  We explore the physics of a Lagrangian fixed point within the framework of a
gravitational average effective action with scale-dependent couplings. A
concrete model with a Lagrangian fixed point in four-dimensional space-time is
formulated. The cosmological equations of this model are then solved
analytically. The solution offers several non-trivial branches, which have the
characteristics of a bouncing cosmology with late time inflation and variable
gravitational couplings.
