Pisot substitutions and Rauzy fractals

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