  We have investigated the three-color Ashkin-Teller model (3AT), on the
Wheatstone bridge hierarchical lattice, by means of a Migdal-Kadanoff
renormalization group approach. We have obtained the exact recursion relations
for the renormalized couplings, which have been used to investigate the phase
diagram and to study the corresponding critical points. The phase diagram,
represented in terms of the dual transmissivity vector, presents four magnetic
phases and nine critical points. We have also numerically calculated the
correlation length ($\nu_T$) and crossover ($\phi$) critical exponents, which
show that seven of the critical points are in the Potts model universality
class ($q=2$, 4 e 8). The remaining critical points are in a universality class
which may belong to the Baxter's line. Our results are exact on the
hierarchical lattice used in the present work and the phase diagram can be
considered as an approximation to more realistic Bravais lattices.
