Download this article
Download this article For screen
For printing
Recent Issues
Volume 12, Issue 3
Volume 12, Issue 2
Volume 12, Issue 1
Volume 11, Issue 4
Volume 11, Issue 3
Volume 11, Issue 2
Volume 11, Issue 1
Volume 10, Issue 4
Volume 10, Issue 3
Volume 10, Issue 2
Volume 10, Issue 1
Volume 9, Issue 4
Volume 9, Issue 3
Volume 9, Issue 2
Volume 9, Issue 1
Volume 8, Issue 4
Volume 8, Issue 3
Volume 8, Issue 2
Volume 8, Issue 1
Older Issues
Volume 7, Issue 4
Volume 7, Issue 3
Volume 7, Issue 2
Volume 7, Issue 1
Volume 6, Issue 4
Volume 6, Issue 2-3
Volume 6, Issue 1
Volume 5, Issue 4
Volume 5, Issue 3
Volume 5, Issue 1-2
Volume 4, Issue 4
Volume 4, Issue 3
Volume 4, Issue 2
Volume 4, Issue 1
Volume 3, Issue 3-4
Volume 3, Issue 2
Volume 3, Issue 1
Volume 2, Issue 4
Volume 2, Issue 3
Volume 2, Issue 2
Volume 2, Issue 1
Volume 1, Issue 4
Volume 1, Issue 3
Volume 1, Issue 2
Volume 1, Issue 1
The Journal
About the Journal
Editorial Board
Subscriptions
 
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
 
founded and published with the
scientific support and advice of
mathematicians from the
Moscow Institute of
Physics and Technology
 
ISSN (electronic): 2640-7361
ISSN (print): 2220-5438
Author Index
To Appear
 
Other MSP Journals
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
Abstract

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.

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