Overlap coincidence to strong coincidence in substitution tiling dynamics

Overlap coincidence is an equivalent criterion to pure discrete spectrum of the dynamics of self-affine tilings in R. In the case of d = 1, strong coincidence on m-letter irreducible substitution has been introduced in [10, 8] which implies that the system is metrically conjugate to a domain exchange in Rm−1. However being a domain exchange does not imply the property of pure discrete spectrum of the tiling dynamics. The relation between two coincidences has not been established completely. In this paper we generalize strong coincidence to higher dimensions and show the implication from overlap coincidence to the new strong coincidence when the associated height group is trivial. Furthermore we introduce a new criterion ‘simultaneous coincidence’ and show the implication from overlap coincidence to the simultaneous coincidence. The triviality of height group is shown in [6, 18] for 1-dimension irreducible Pisot substitutions.