Coincidence for substitutions of Pisot type

Abstract

Abstract. -Let ϕ be a substitution of Pisot type on the alphabet A = {1, 2, . . . , d}; ϕ satisfies the strong coincidence condition if for every i, j ∈ A, there are integers k, n such that ϕ n (i) and ϕ n (j) have the same k-th letter, and the prefixes of length k − 1 of ϕ n (i) and ϕ n (j) have the same image under the abelianization map. We prove that the strong coincidence condition is satisfied if d = 2 and provide a partial result for d ≥ 2.Résumé (Coïncidence pour les substitutions de type Pisot). -Soit ϕ une substitution de type Pisot sur un alphabet A = {1, 2, . . . , d} ; on dit que ϕ satisfait la condition de coïncidence forte si pour tout i, j ∈ A, il existe des entiers k, n tels que ϕ n (i) et ϕ n (j) aient la même k-ième lettre et les préfixes de longueur k − 1 de ϕ n (i) et ϕ n (j) aient la même image par l'application d'abélianisation. Nous montrons que la condition de coïncidence forte est satisfaite pour d = 2 et nous donnons un résultat partiel pour d ≥ 2.

Fractal representation of the attractive lamination of an automorphism of the free group

“…To ensure that the subtiles are disjoint in measure, we recall from [9] the following combinatorial condition on substitutions. This condition is satisfied by every unimodular irreducible Pisot substitution over a two-letter alphabet [10]. It is conjectured that every substitution of Pisot type satisfies the strong coincidence condition.…”

Balanced pair algorithm for a class of cubic substitutions

“…For references on conditions under which the Pisot conjecture is true, we refer to, among other references, [1,2,3,4,5,6,7,11,12,15,16,21,22,23].…”

“…Many classes of substitutions are shown to satisfy this condition. For example, Barge and Diamond proved in [4] that every irreducible Pisot substitution over 2 letters satisfies it. It is conjectured that this is true for alphabets of arbitrary size but a general proof is still outstanding.…”

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“…The inflation and substitution map F σ is defined as follows on the set of subsets of G ( [8]). Define…”