YET ANOTHER CHARACTERIZATION OF THE PISOT SUBSTITUTION CONJECTURE

Encore une caractérisation de la conjecture Pisot pour les substitutions

Paul Mercat (1) , Shigeki Akiyama (2)

1 I2M - Institut de Mathématiques de Marseille
2 Department of Mathematics

We give a sufficient geometric condition for a subshift to be measurably isomorphic to a domain exchange and to a translation on a torus. And for an irreducible unit Pisot substitution, we introduce a new topology on the discrete line and give a simple necessary and sufficient condition for the symbolic system to have pure discrete spectrum. This condition gives rise to an algorithm based on computation of automata. To see the power of this criterion, we provide families of substitutions that satisfies the Pisot substitution conjecture : 1) a → a^kbc, b → c, c → a, for k ∈ N and 2) a → a^lba^(k−l), b → c, c → a, for k ∈ N≥1, for 0 ≤ l ≤ k using different methods. And we also provide an example of S-adic system with pure discrete spectrum everywhere.

Mots clés: Substitution; Symbolic dynamics; Fractal; Pisot Number

Domaines: Systèmes dynamiques [math.DS]; Mathématique discrète [cs.DM]; Géométrie métrique [math.MG]; Mathématiques générales [math.GM]; Géométrie algorithmique [cs.CG]; Théorie et langage formel [cs.FL]; Théorie des nombres [math.NT]

Paul Mercat, Shigeki Akiyama. YET ANOTHER CHARACTERIZATION OF THE PISOT SUBSTITUTION CONJECTURE. 2018. ⟨hal-01781126v1⟩