A combinatorial approach to products of Pisot substitutions

Valérie Berthé (1) , Jérémie Bourdon (2) , Timo Jolivet (3) , Anne Siegel (4)

1 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications
2 LINA - Laboratoire d'Informatique de Nantes Atlantique
3 IMT - Institut de Mathématiques de Toulouse UMR5219
4 Dyliss - Dynamics, Logics and Inference for biological Systems and Sequences

We define a generic algorithmic framework to prove a pure discrete spectrum for the substitutive symbolic dynamical systems associated with some infinite families of Pisot substitutions. We focus on the families obtained as finite products of the three-letter substitutions associated with the multidimensional continued fraction algorithms of Brun and Jacobi–Perron. Our tools consist in a reformulation of some combinatorial criteria (coincidence conditions), in terms of properties of discrete plane generation using multidimensional (dual) substitutions. We also deduce some topological and dynamical properties of the Rauzy fractals, of the underlying symbolic dynamical systems, as well as some number-theoretical properties of the associated Pisot numbers.

Domaines: Combinatoire [math.CO]; Systèmes dynamiques [math.DS]

Valérie Berthé, Jérémie Bourdon, Timo Jolivet, Anne Siegel. A combinatorial approach to products of Pisot substitutions. Ergodic Theory and Dynamical Systems, 2016, 36 (6), pp.1757-1794. ⟨10.1017/etds.2014.141⟩. ⟨hal-01196326⟩