Pisot substitutions and Rauzy fractals

We prove that the dynamical system generated by a primitive unimodular substitution of the Pisot type on d letters satisfying a combinatorial condition which is easy to check, is measurably isomorphic to a domain `exchange in Rd-1, and is a finite extension of a translation on the torus Td-1. I, the course of the proof, we introduce some potentially useful notions: the linear maps associated to a substitution and their dual maps, and the sigma -structure for a dynamical system with respect to a pair of partitions.