  For scalar QED on a three-dimensional toroidal lattice with a fine lattice
spacing we consider the renormalization problem of choosing counter terms
depending on the lattice spacing, so that the theory stays finite as the
spacing goes to zero. We employ a renormalization group method which analyzes
the flow of the mass and the vacuum energy as a problem in discrete dynamical
systems. The main result is that counter terms can be choosen so that at the
end of the iteration these quantities take preassigned values. No use is made
of perturbation theory. The renormalization group transformations are defined
with bounded fields, an approximation which can be justified in Balaban's
approach to the renormalization group.
