Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-03T00:38:10.587Z Has data issue: false hasContentIssue false
  • English
  • Français

Equidistribution of dense subgroups on nilpotent Lie groups

Published online by Cambridge University Press:  23 June 2009

EMMANUEL BREUILLARD
EMMANUEL BREUILLARD*
Affiliation:
Ecole Polytechnique, 91128 Palaiseau, France (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Let Γ be a dense subgroup of a simply connected nilpotent Lie group G generated by a finite symmetric set S. We consider the n-ball Sn for the word metric induced by S on Γ. We show that Sn (with uniform measure) becomes equidistributed on G with respect to the Haar measure as n tends to infinity. We also prove the analogous result for random walk averages.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 1 , February 2010 , pp. 131 - 150
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Alexopoulos, G.. Random walks on discrete groups of polynomial volume growth. Ann. Probab. 30(2) (2002), 723–801.CrossRefGoogle Scholar
[2]Arnol’d, V. I. and Krylov, A. L.. Uniform distribution of points on a sphere and certain ergodic properties of solutions of linear ordinary differential equations in a complex domain. Dokl. Akad. Nauk SSSR 148 (1963), 9–12.Google Scholar
[3]Auslander, L. and Brezin, J.. Uniform distribution in solvmanifolds. Adv. Math. 7 (1971), 111–144.CrossRefGoogle Scholar
[4]Babillot, M.. Points entiers et groupes discrets, de l’analyse aux systèmes dynamiques. Rigidité, groupe fondamental et dynamique (Panoramas et Synthèses SMF Monographs, 13). Société Mathématique de France, 2002, pp. 1–119.Google Scholar
[5]Bass, H.. The degree of polynomial growth of finitely generated nilpotent groups. Proc. London Math. Soc. 25(3) (1972), 603–614.CrossRefGoogle Scholar
[6]Bellaiche, A.. The tangent space in sub-Riemannian geometry. Sub-Riemannian Geometry (Progress of Mathematics, 144). Eds. A. Bellaiche and J.-J. Risler. Birkhäuser, Basel, 1996, pp. 1–78.CrossRefGoogle Scholar
[7]Breuillard, E.. Random walks on Lie groups, survey. Preprint, http://www.math.polytechnique/∼breuilla/part0gb.pdf.Google Scholar
[8]Breuillard, E.. Geometry of locally compact groups with polynomial growth and shape of large balls. Preprint, 2007, arXiv:0704.0095.Google Scholar
[9]Breuillard, E.. Local limit theorems and equidistribution of random walks on the Heisenberg group. Geom. Funct. Anal. (GAFA) 15(1) (2005), 49.Google Scholar
[10]Duke, W., Rudnick, N. and Sarnak, P.. Density of integer points on affine homogeneous varieties. Duke Math. J. 71(1) (1993), 143–179.CrossRefGoogle Scholar
[11]Eskin, A.. Counting problems and semisimple groups. Proc. of the Int. Congress of Mathematicians. Vol. II (Berlin, 1998). Doc. Math. 1998, Extra Vol. II, 539–552.Google Scholar
[12]Eskin, A., Mozes, S. and Shah, N.. Unipotent flows and counting lattice points on homogeneous varieties. Ann. of Math. (2) 143 (1996), 253–299.CrossRefGoogle Scholar
[13]Goodman, R. W.. Nilpotent Lie Groups: Structure and Applications to Analysis (Lectures Notes in Mathematics, 562). Springer, Berlin, 1976.CrossRefGoogle Scholar
[14]Gorodnik, A.. Lattice action on the boundary of . Ergod. Th. & Dynam. Sys. 23(6) (2003), 1817–1837.CrossRefGoogle Scholar
[15]Gromov, M.. Carnot–Carathéodory spaces seen from within. Sub-Riemannian Geometry. Eds. A. Bellaiche and J.-J. Risler. Birkäuser, Basel, 1996, pp. 79–323.CrossRefGoogle Scholar
[16]Guivarc’h, Y.. Croissance polynomiale et périodes des fonctions harmoniques. Bull. Soc. Math. France 101 (1973), 353–379.Google Scholar
[17]Guivarc’h, Y.. Equirépartition dans les espaces homogènes. Théorie ergodique (Actes Journées Ergodiques, Rennes, 1973/1974) (Lecture Notes in Mathematics, 532). Springer, Berlin, 1976,pp. 131–142.CrossRefGoogle Scholar
[18]Kazhdan, D. A.. Uniform distribution on a plane. Trudy Moskov. Mat. Ob. 14 (1965), 299–305.Google Scholar
[19]Ledrappier, F.. Ergodic Properties of some linear actions. J. Math. Sci. 105(2) (2001).CrossRefGoogle Scholar
[20]Le Page, E.. Théorèmes quotients pour certaines marches aléatoires. C. R. Acad. Sci. série A 279(2) (1974).Google Scholar
[21]Pansu, P.. Croissance des boules et des géodé siques fermées dans les nilvariétés. Ergod. Th. & Dynam. Sys. 3(3) (1983), 415–445.CrossRefGoogle Scholar
[22]Raghunathan, M. S.. Discrete Subgroups of Lie Groups. Springer, Berlin, 1972.CrossRefGoogle Scholar
[23]Starkov, A.. Dynamical Systems on Homogeneous Spaces (Translations of Mathematical Monographs, 190). Americal Mathematical Society, Providence, RI, 2000.CrossRefGoogle Scholar
[24]Weyl, H.. Über die gleichverteilung von Zahlen mod. Eins. Math. Ann. 77 (1916), 313–352.CrossRefGoogle Scholar