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A brief remark on orbits of SL(2,ℤ) in the Euclidean plane

Published online by Cambridge University Press:  21 July 2009

ANTONIN GUILLOUX
ANTONIN GUILLOUX*
Affiliation:
Université de Lyon, ENS Lyon, U.M.P.A., 46, Allée d’Italie, 69007 Lyon, France (email: [email protected])
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Abstract

The repartition of dense orbits of lattices in the Euclidean plane were described by Ledrappier and Nogueira. We present here an elementary description of the gaps appearing in the experimentations. The main idea behind this description is to see the Euclidean plane as the space of (upper triangular) unipotent orbits in SL(2,ℝ). We conclude with the remark that this analysis may be carried on in much more general settings.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 4 , August 2010 , pp. 1101 - 1109
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Borel, A.. Linear Algebraic Groups, 2nd edn(Graduate Texts in Mathematics, 126). Springer, New York, 1991.CrossRefGoogle Scholar
[2]Gorodnik, A.. Uniform distribution of orbits of lattices on spaces of frames. Duke Math. J. 122(3) (2004), 549–489CrossRefGoogle Scholar
[3]Gorodnik, A. and Weiss, B.. Distribution of lattice orbits on homogeneous varieties. Geom. Funct. Anal. 17(1) (2007), 58–115.CrossRefGoogle Scholar
[4]Guilloux, A.. Polynomial dynamic and lattice orbits in S-arithmetic homogeneous spaces, Preprint, arXiv: 0902:1910, http://www.umpa.ens-lyon.fr/∼aguillou/articles/equihomo.pdf.Google Scholar
[5]Ledrappier, F.. Distribution des orbites des réseaux sur le plan réel. C. R. Math. Acad. Sci. Paris 329 (1999), 61–64.CrossRefGoogle Scholar
[6]Ledrappier, F. and Pollicott, M.. Distribution results for lattices in SL(2,ℚp). Bull. Braz. Math. Soc. (N.S.) 36(2) (2005), 143–176.CrossRefGoogle Scholar
[7]Nogueira, A.. Orbit distribution on ℝ2 under the natural action of SL(2,ℤ). Indag. Math. (N.S.) 13(1) (2002), 103–124.CrossRefGoogle Scholar