Invariance properties of Sturmian words
type: Item
instance of: scholarly article
title: Invariance properties of Sturmian words
zbMATH Open document ID: 0904.11008
author: Bruno Parvaix
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1999-01-19T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/231574
full work available at URL: jtnb9-2.html
review text: In this interesting paper the author studies Sturmian (i.e., billiard) binary sequences that are invariant under a non-trivial substitution. The case of a trajectory going through the origin has been first addressed by \textit{D. Crisp, W. Moran, A. Pollington} and \textit{P. Shiue} [J. Théor. Nombres Bordx. 5, 123-138 (1993; Zbl 0786.11041); see also \textit{J. Berstel} and \textit{P. Séébold}, Bull. Belg. Math. Soc. -- Simon Stevin 1, 175-189 (1994; Zbl 0803.68095) and \textit{T. Komatsu} and \textit{A. J. van der Poorten}, Jap. J. Math., New Ser. 22, 349-354 (1996; Zbl 0868.11015)]. In the paper under review, the author calls ``Sturm numbers'' the slopes pointed out in the paper of Crisp, Moran, Pollington and Shiue. He proves that a necessary condition for any Sturmian sequence to be fixed by a non-trivial substitution is that the slope be a Sturm number. The condition is proved sufficient for a class of intercepts that are homographic functions of the slope.
Mathematics Subject Classification ID: 11B85
Mathematics Subject Classification ID: 68R15
zbMATH DE Number: 1164092
zbMATH Keywords: Sturmian binary sequences
zbMATH Keywords: Sturm numbers
zbMATH Keywords: substitutions
zbMATH Keywords: square billiards
reviewed by: Jean-Paul Allouche
MaRDI profile type: Publication
OpenAlex ID: W2312245955
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DOI: 10.5802/JTNB.207
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description: scientific article; zbMATH DE number 1164092