  If phi is a Pisot substitution of degree d, then the inflation and
substitution homeomorphism Phi on the tiling space T_Phi factors via geometric
realization onto a d-dimensional solenoid. Under this realization, the
collection of Phi-periodic asymptotic tilings corresponds to a finite set that
projects onto the branch locus in a d-torus. We prove that if two such tiling
spaces are homeomorphic, then the resulting branch loci are the same up to the
action of certain affine maps on the torus.
