Orbit distribution on \(\mathbb R^2\) under the natural action of \(\mathrm{SL}(2,\mathbb Z)\)
type: Item
instance of: scholarly article
title: Orbit distribution on \(\mathbb R^2\) under the natural action of \(\mathrm{SL}(2,\mathbb Z)\)
zbMATH Open document ID: 1016.37003
DOI: 10.1016/S0019-3577(02)90009-1
author: Arnaldo Nogueira
published in: Indagationes Mathematicae. New Series
publication date: 2003-06-05T00:00:00Z
review text: The distribution of orbits under the natural linear action of \(\mathrm{SL}(2,\mathbb Z\) on the plane is studied. An application/example is the evaluation of the asymptotic behaviour of the set \(\{A\in \mathrm{SL}(2,\mathbb Z)\mid Ax\in [0,r]^2, \|A\|\leq t\}\) for large \(r,t>0\), where \(x\) is an explicit vector with incommensurable entries. More general results in this direction have been obtained by \textit{F. Ledrappier} [C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 1, 61--64 (1999; Zbl 0928.22012)] and others using the ergodic theory of Lie group actions. In this paper more restricted results are reached using number theory methods, giving sharper and more explicit error terms.
reviewed by: Thomas B. Ward
Mathematics Subject Classification ID: 37A15
Mathematics Subject Classification ID: 22D40
zbMATH DE Number: 1893633
zbMATH Keywords: linear actions
zbMATH Keywords: orbit distribution
zbMATH Keywords: ergodic theory
zbMATH Keywords: Lie group actions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1893633