Discrete planes, \({\mathbb Z}^2\)-actions, Jacobi-Perron algorithm and substitutions
type: Item
instance of: scholarly article
title: Discrete planes, \({\mathbb Z}^2\)-actions, Jacobi-Perron algorithm and substitutions
zbMATH Open document ID: 1017.11006
author: Pierre Arnoux
author: Valérie Berthé
author: Shunji Ito
publication date: 2002-09-03T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/115982
review text: This paper discusses an explicit method to build a discrete approximation of an irrational plane in \(\mathbb R^3\). This approximation can be described by a two-dimensional sequence, which is directly related to symbolic dynamics for a \(\mathbb Z^2\)-action by rotations on the unit circle. This sequence can be generated by applying the Jacobi-Perron algorithm to the coordinates of the unit vector orthogonal to the given plane. This paper attempts to generalize to higher dimensions well-known results for the usual continued fractions.
Mathematics Subject Classification ID: 11A55
Mathematics Subject Classification ID: 11J70
Mathematics Subject Classification ID: 40A15
Mathematics Subject Classification ID: 68R15
zbMATH DE Number: 1794801
zbMATH Keywords: substitutions
zbMATH Keywords: generalized continued fractions
zbMATH Keywords: discrete planes
zbMATH Keywords: tilings
zbMATH Keywords: Jacobi-Perron algorithm
zbMATH Keywords: induction
zbMATH Keywords: \(\mathbb Z^2\)-actions
zbMATH Keywords: two-dimensional sequences
reviewed by: Takao Komatsu
MaRDI profile type: Publication
OpenAlex ID: W161743208
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DOI: 10.5802/AIF.1889
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published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1794801