The Pisot conjecture for -substitutions
MARCY BARGE · Ergodic Theory and Dynamical Systems · 2016
We prove the Pisot conjecture for $\unicode[STIX]{x1D6FD}$ -substitutions: if $\unicode[STIX]{x1D6FD}$ is a Pisot number, then the tiling dynamical system $(\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D6FD}}},\mathbb{R})$ associated with the $\unicode[STIX]{x1D6FD}$ -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.