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On Arnol’d’s and Kazhdan’s equidistribution problems

Published online by Cambridge University Press:  23 November 2011

ALEXANDER GORODNIK  and
AMOS NEVO
ALEXANDER GORODNIK
Affiliation:
School of Mathematics, University of Bristol, Bristol, UK (email: a.gorodnik@bristol.ac.uk)
AMOS NEVO
Affiliation:
Department of Mathematics, Technion, Haifa, Israel (email: anevo@tx.technion.ac.il)
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Abstract

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We consider isometric actions of lattices in semisimple algebraic groups on (possibly non-compact) homogeneous spaces with (possibly infinite) invariant Radon measure. We assume that the action has a dense orbit, and demonstrate two novel and non-classical dynamical phenomena that arise in this context. The first is the existence of a mean ergodic theorem even when the invariant measure is infinite; this implies the existence of an associated limiting distribution, which can be different from the underlying invariant measure. The second is uniform quantitative equidistribution of all orbits in the space, which follows from a quantitative mean ergodic theorem for such actions. In turn, these results imply quantitative ratio ergodic theorems for isometric actions of lattices. This sheds some unexpected light on certain equidistribution problems posed by Arnol’d [Arnol’d’s Problems. Springer, Berlin, 2004] and also on the ratio equidistribution conjecture for dense subgroups of isometries formulated by Kazhdan [Uniform distribution on a plane. Tr. Mosk. Mat. Obs. 14 (1965), 299–305]. We briefly mention the general problem regarding ergodic theorems for actions of lattices on homogeneous spaces and its solution given by Gorodnik and Nevo [Duality principle and ergodic theorems, in preparation], and present a number of examples to demonstrate our results. Finally, we also prove results on quantitative equidistribution for absolutely continuous averages in transitive actions.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 6 , December 2012 , pp. 1972 - 1990
Copyright
Copyright © Cambridge University Press 2011

References

[Aa]Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50). American Mathematical Society, Providence, RI, 1997.CrossRefGoogle Scholar
[A]Arnol’d, V.. Arnold’s Problems. Springer, Berlin, 2004.Google Scholar
[AK]Arnol’d, V. and Krylov, A.. 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
[B]Bewley, T.. Sur l’application des théorèmes ergodiques aux groupes libres de transformations: un contre-exemple. C. R. Acad. Sci. Paris Sér. A–B 270 (1970), A1533–A1534.Google Scholar
[BG]Bourgain, J. and Gamburd, A.. On the spectral gap for finitely-generated subgroups of SU(2). Invent. Math. 171(1) (2008), 83–121.CrossRefGoogle Scholar
[Br1]Breuillard, E.. Equidistribution of random walks on nilpotent Lie groups and homogeneous spaces, chapter 1: Random walks on Lie groups. PhD Thesis, Department of Mathematics, Yale University, 2004.Google Scholar
[Br2]Breuillard, E.. Equidistribution of dense subgroups on nilpotent Lie groups. Ergod. Th. & Dynam. Sys. 30 (2010), 131–150.CrossRefGoogle Scholar
[C]Clozel, L.. Automorphic forms and the distribution of points on odd-dimensional spheres. Israel J. Math. 132 (2002), 175–187.CrossRefGoogle Scholar
[CO]Clozel, L. and Otal, J.-P.. Unique ergodicité des correspondances modulaires. Essays on Geometry and Related Topics (Monographs de L’Enseignement Mathématique, 38). Enseignement Mathématique, Geneva, 2001, pp. 205–216.Google Scholar
[CU]Clozel, L. and Ullmo, E.. Équidistribution des points de Hecke. Contributions to Automorphic Forms, Geometry, and Number Theory. Johns Hopkins University Press, Baltimore, MD, 2004, pp. 193–254.Google Scholar
[Do]Dolgopyat, D.. On mixing properties of compact group extensions of hyperbolic systems. Israel J. Math. 130 (2002), 157–205.CrossRefGoogle Scholar
[G1]Gorodnik, A.. Lattice action on the boundary of ${\rm SL}(n,\Bbb R)$. Ergod. Th. & Dynam. Sys. 23(6) (2003), 1817–1837.CrossRefGoogle Scholar
[G2]Gorodnik, A.. Uniform distribution of orbits of lattices on spaces of frames. Duke Math. J. 122(3) (2004), 549–589.CrossRefGoogle Scholar
[GM]Gorodnik, A. and Maucourant, F.. Proximality and equidistribution on the Furstenberg boundary. Geom. Dedicata 113 (2005), 197–213.CrossRefGoogle Scholar
[GN1]Gorodnik, A. and Nevo, A.. The Ergodic Theory of Lattice Subgroups (Annals of Mathematics Studies, 172). Princeton University Press, Princeton, NJ, 2010.Google Scholar
[GN2]Gorodnik, A. and Nevo, A.. Duality principle and ergodic theorems, in preparation.Google Scholar
[GO]Gorodnik, A. and Oh, H.. Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary. Duke Math. J. 139(3) (2007), 483–525.CrossRefGoogle Scholar
[GW]Gorodnik, A. and Weiss, B.. Distribution of lattice orbits on homogeneous varieties. Geom. Funct. Anal. 17(1) (2007), 58–115.CrossRefGoogle Scholar
[Gu1]Guivarc’h, Y.. Généralisation d’un théorème de von Neumann. C. R. Acad. Sci. Paris 268 (1969), 1020–1023.Google Scholar
[Gu2]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
[HP]Haagerup, U. and Przybyszewska, A.. Proper metrics on locally compact groups, and proper affine isometric actions on Banach spaces. Preprint, 2006, arXiv:math/0606794v1.Google Scholar
[HR]Hewitt, E. and Ross, K.. Abstract Harmonic Analysis. Vol. I, 2nd edn.(Grundlehren der Mathematischen Wissenschaften, 115). Springer, Berlin, 1979.CrossRefGoogle Scholar
[K]Kazhdan, D.. Uniform distribution on a plane. Tr. Mosk. Mat. Obs. 14 (1965), 299–305.Google Scholar
[L]Ledrappier, F.. Distribution des orbites des réseaux sur le plan réel. C. R. Acad. Sci. Paris Sér. I Math. 329(1) (1999), 61–64.CrossRefGoogle Scholar
[LP1]Ledrappier, F. and Pollicott, M.. Ergodic properties of linear actions of (2×2)-matrices. Duke Math. J. 116(2) (2003), 353–388.CrossRefGoogle Scholar
[LP2]Ledrappier, F. and Pollicott, M.. Distribution results for lattices in ${\rm SL}(2,\Bbb Q_p)$. Bull. Braz. Math. Soc. (N.S.) 36(2) (2005), 143–176.CrossRefGoogle Scholar
[LPS1]Lubotzky, A., Phillips, R. and Sarnak, P.. Hecke operators and distributing points on the sphere. I.. Comm. Pure Appl. Math. 39(suppl.) (1986), S149–S186, Proceedings of the Symposium on Frontiers of the Mathematical Sciences (New York, 1985).CrossRefGoogle Scholar
[LPS2]Lubotzky, A., Phillips, R. and Sarnak, P.. Hecke operators and distributing points on S 2. II. Comm. Pure Appl. Math. 40(4) (1987), 401–420.CrossRefGoogle Scholar
[MNS]Margulis, G., Nevo, A. and Stein, E.. Analogs of Wiener’s ergodic theorems for semisimple Lie groups. II. Duke Math. J. 103(2) (2000), 233–259.CrossRefGoogle Scholar
[Ne1]Nevo, A.. Harmonic analysis and pointwise ergodic theorems for non-commuting transformations. J. Amer. Math. Soc. 7(4) (1994), 875–902.CrossRefGoogle Scholar
[Ne2]Nevo, A.. Spectral transfer and pointwise ergodic theorems for semi-simple Kazhdan groups. Math. Res. Lett. 5(3) (1998), 305–325.CrossRefGoogle Scholar
[Ne3]Nevo, A.. Pointwise ergodic theorems for actions of groups. Handbook of Dynamical Systems, Vol. 1B. Elsevier, Amsterdam, 2006, pp. 871–982.Google Scholar
[No]Nogueira, A.. Orbit distribution on $\Bbb R^2$ under the natural action of ${\rm SL}(2,\Bbb Z)$. Indag. Math. (N.S.) 13(1) (2002), 103–124.CrossRefGoogle Scholar
[O]Oh, H.. The Ruziewicz problem and distributing points on homogeneous spaces of a compact Lie group. Israel J. Math. 149 (2005), 301–316.CrossRefGoogle Scholar
[V1]Vorobets, Y.. On the uniform distribution of orbits of finitely generated groups and semigroups of plane isometries. Mat. Sb. 195(2) (2004), 17–40; Engl. transl. Sb. Math. 195(1–2) (2004), 163–186.Google Scholar
[V2]Vorobets, Y.. On the actions of finitely generated groups and semigroups on a plane by means of isometries. Mat. Zametki 75(4) (2004), 523–548; Engl. transl. Math. Notes 75(3–4) (2004), 489–512.Google Scholar