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 ($ u_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 Baxters 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.
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