Representation of non-unimodular substitution dynamical systems

The combinatorial condition of strong coincidence is proved to be a sufficient condition for the dynamical system associated with a non-unimodular substitution of Pisot type to be measure-theoretic ally isomorphic with an exchange of domain on a set called the Rauzy fractal of the substitution. The Rauzy fractal is a self-similar compact subset of the product of an Euclidean space with finite extensions of p-adic fields. As a consequence, every substitutive dynamical system of Pisot type is a finite extension of its maximal equicontinuous factor. We prove that this maximal factor contains a p-adic translation if and only if the incidence matrix of the substitution is nilpotent modulo p.