Download this article
Download this article For screen
For printing
Recent Issues
Volume 13, Issue 2
Volume 13, Issue 1
Volume 12, Issue 4
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
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
founded and published with the
scientific support and advice of
mathematicians from the
Moscow Institute of
Physics and Technology
Subscriptions
 
ISSN (electronic): 2996-220X
ISSN (print): 2996-2196
Author Index
To Appear
 
Other MSP Journals
This article is available for purchase or by subscription. See below.
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.

PDF Access Denied

We have not been able to recognize your IP address 100.26.176.111 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

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