Coincidence for substitutions of Pisot type. (English) Zbl 1028.37008

Barge, Marcy; Diamond, Beverly

Bull. Soc. Math. Fr. 130, No. 4, 619-626 (2002).

A substitution \(\sigma\) on a finite alphabet satisfies the strong coincidence condition if for every letters \(i\) and \(j\) of the alphabet, there exist integers \(n, k\) such that \(\sigma^n(i)\) and \(\sigma^n(k)\) have the same \(k\)th letter, and their prefixes of length \(k-1\) contain the same number of occurrence of each letter. In the case in which the dominant eigenvalue of the transition matrix of the substitution is a Pisot number, the authors prove that all substitutions on two letters satisfy the strong coincidence condition, and obtain a partial result for substitutions on more than two letters.

Reviewer: Franco Vivaldi (London)

MSC:

37B10

Symbolic dynamics

Keywords:

substitutions; dynamical system; Pisot number; coincidence conjecture; pure discrete spectrum