The Pisot conjecture for -substitutions

We prove the Pisot conjecture for -substitutions: if is a Pisot number, then the tiling dynamical system associated with the -substitution has pure discrete spectrum. As corollaries: (1) arithmetical coding of the hyperbolic solenoidal automorphism associated with the companion matrix of the minimal polynomial of any Pisot number is almost everywhere one-to-one; and (2) all Pisot numbers are weakly finitary.