  We prove that the generator of the renormalization group of Potts models on
hierarchical lattices can be represented by a rational map acting on a
finite-dimensional product of complex projective spaces. In this framework we
can also consider models with an applied external magnetic field and
multiple-spin interactions. We use recent results regarding iteration of
rational maps in several complex variables to show that, for some class of
hierarchical lattices, Lee-Yang and Fisher zeros belong to the unstable set of
the renormalization map.
