Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-04-30T20:10:25.725Z Has data issue: false hasContentIssue false
  • English
  • Français

A word of low complexity without uniform frequencies

Part of: Discrete mathematics in relation to computer science Topological dynamics Ergodic theory

Published online by Cambridge University Press:  29 May 2023

JULIEN CASSAIGNE  and
IDRISSA KABORÉ Open the ORCID record for IDRISSA KABORÉ [Opens in a new window]
JULIEN CASSAIGNE
Affiliation:
Institut de Mathématiques de Marseille, 163 avenue de Luminy, case 907, F-13288 Marseille Cedex 9, France (e-mail: julien.cassaigne@math.cnrs.fr)
IDRISSA KABORÉ*
Affiliation:
UFR-Sciences Exactes et Appliquées, Université Nazi Boni, 01 BP 1091 Bobo-Dioulasso 01, Burkina Faso
*
e-mail: ikaborei@yahoo.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we construct a uniformly recurrent infinite word of low complexity without uniform frequencies of letters. This shows the optimality of a bound of Boshernitzan, which gives a sufficient condition for a uniformly recurrent infinite word to admit uniform frequencies.

Keywords

infinite wordsubstitutioncomplexityuniform frequencies

MSC classification

Primary: 37B10: Symbolic dynamics 37A25: Ergodicity, mixing, rates of mixing 68R15: Combinatorics on words
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 4 , April 2024 , pp. 1013 - 1025
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Berthé, V. and Rigo, M. (Eds). Combinatorics, Automata and Number Theory (Encyclopedia of Mathematics and its Applications, 135). Cambridge University Press, Cambridge, 2010.Google Scholar
Boshernitzan, M.. A unique ergodicity of minimal symbolic flows with minimal block growth. J. Anal. Math. 44 (1985), 77–96.Google Scholar
Cassaigne, J.. Complexité et facteurs spéciaux. Bull. Belg. Math. Soc. 4 (1997), 67–88.Google Scholar
Cyr, V. and Kra, B.. Counting generic measures for a subshift of linear growth. J. Eur. Math. Soc. (JEMS) 21 (2019), 355–380.Google Scholar
Cyr, V. and Kra, B.. Realizing ergodic properties in zero entropy subshifts. Israel J. Math. 240 (2020), 119–148.CrossRefGoogle Scholar
Damron, M. and Fickenscher, J.. The number of ergodic measures for transitive subshifts under the regular bispecial condition. Ergod. Th. & Dynam. Sys. 42 (2022), 86–140.Google Scholar
Keane, M.. Non-ergodic interval exchange transformations. Israel J. Math. 26 (1977), 188–196.Google Scholar
Monteil, T.. Illumination dans les billards polygonaux et dynamique symbolique. PhD Thesis, Institut de Mathématiques de Luminy, Université de la Méditerranée, 2005.Google Scholar