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Almost everywhere balanced sequences of complexity $2n+1$

Julien Cassaigne, Sébastien Labbé and Julien Leroy

Vol. 11 (2022), No. 4, 287–333
DOI: 10.2140/moscow.2022.11.287

We study ternary sequences associated with a multidimensional continued fraction algorithm introduced by the first author. The algorithm is defined by two matrices and we show that it is measurably isomorphic to the shift on the set {1,2} of directive sequences. For a given set 𝒞 of two substitutions, we show that there exists a 𝒞-adic sequence for every vector of letter frequencies or, equivalently, for every directive sequence. We show that their factor complexity is at most 2n + 1 and is equal to 2n + 1 if and only if the letter frequencies are rationally independent if and only if the 𝒞-adic representation is primitive. It turns out that in this case, the sequences are dendric. We also prove that μ-almost every 𝒞-adic sequence is balanced, where μ is any shift-invariant ergodic Borel probability measure on {1,2} giving a positive measure to the cylinder [12121212]. We also prove that the second Lyapunov exponent of the matrix cocycle associated with the measure μ is negative.

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substitutions, factor complexity, Selmer algorithm, continued fraction, bispecial, Lyapunov exponents, balance
Mathematical Subject Classification
Primary: 37B10
Secondary: 11J70, 37H15, 68R15
Received: 26 August 2021
Revised: 25 April 2022
Accepted: 9 May 2022
Published: 25 November 2022
Julien Cassaigne
Institut de Mathématiques de Marseille
Aix-Marseille Université, CNRS, Centrale Marseille
12M - UMR 7373
Sébastien Labbé
Laboratoire Bordelais de Recherche en Informatique
Université de Bordeaux
Julien Leroy
Département de Mathématique
Université de Liège