Pure discrete spectrum dynamical system and periodic tiling associated with a substitution

We give a computable sufficient condition for the symbolic dynamical system associated with a substitution of Pisot type to have a pure discrete spectrum. In the unimodular case, this condition is necessary when the substitution has no nontrivial coboundary; it is satisfied if and only if the Rauzy fractal associated with the substitution generates a self-similar periodic tiling. Conditions for connectivity of Rauzy fractals are deduced.