Geometric theory of unimodular Pisot substitutions

We are concerned with the tiling flow T associated to a substitution 0 over a finite alphabet. Our focus is on substitutions that are unimodular Pisot, i.e., their matrix is unimodular and has all eigenvalues strictly inside the unit circle with the exception of the Perron eigenvalue lambda > 1. The motivation is provided by the (still open) conjecture asserting that T has pure discrete spectrum for any such phi. We develop a number of necessary and sufficient conditions for pure discrete spectrum, including: injectivity of the canonical torus map (the geometric realization), Geometric Coincidence Condition, (partial) commutation of T and the dual Rd-1-action, measure and tiling properties of Rauzy fractals, and concrete algorithms. Some of these are original and some have already appeared in the literature-as sufficient conditions only-but they all emerge from a unified approach based on the new device: the strand space F-phi of phi. The proof of the necessity hinges on determination of the discrete spectrum of T as that of the associated Kronecker toral flow.