Codings of rotations and continued fractions
type: Item
instance of: scholarly article
title: Codings of rotations and continued fractions
zbMATH Open document ID: 0921.11015
author: Gilles Didier
published in: Journal of Number Theory
publication date: 1999-09-29T00:00:00Z
review text: It is well-known that the classical continued fraction algorithm describes a generating process by help of morphisms for Sturmian sequences [see for instance \textit{P. Arnoux} and \textit{G. Rauzy}, Bull. Soc. Math. Fr. 119, 199-215 (1991; Zbl 0789.28011)]. The aim of the paper under review is to extend this result to binary codings of rotations: such a sequence is defined as the coding of the orbit under an irrational rotation of a point on the 1-dimensional torus divided into two intervals (in the Sturmian case, the length of one interval equals the rotation angle). If the lengths of both intervals of the partition are larger than the rotation angle, then it is proved that the set of factors of any binary coding (under this partition) is generated by the iteration of four particular morphisms. Moreover, the continued-fraction-like algorithm which describes the iterations ordering is explicitly given. Note that it remains to explore the connections between this algorithm and some other classical algorithms defined as a skew product of the continued fraction transformation, as for instance in [\textit{Y. Hara-Mimachi} and \textit{S. Ito}, Tokyo J. Math. 14, 251-267 (1991; Zbl 0751.11034)].
Mathematics Subject Classification ID: 11B85
Mathematics Subject Classification ID: 68R15
Mathematics Subject Classification ID: 37E99
Mathematics Subject Classification ID: 11K50
zbMATH DE Number: 1236184
zbMATH Keywords: combinatorics on words
zbMATH Keywords: substitutions
zbMATH Keywords: morphisms for Sturmian sequences
zbMATH Keywords: binary codings of rotations
zbMATH Keywords: continued-fraction-like algorithm
reviewed by: Valérie Berthé
MaRDI profile type: Publication
full work available at URL: jnth.1998.2246
OpenAlex ID: W2039026476
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DOI: 10.1006/JNTH.1998.2246
description: scientific article; zbMATH DE number 1236184