Comptes Rendus
Group Theory
Invariant measures and stiffness for non-Abelian groups of toral automorphisms
Comptes Rendus. Mathématique, Volume 344 (2007) no. 12, pp. 737-742.

Let Γ be a non-elementary subgroup of SL2(Z). If μ is a probability measure on T2 which is Γ-invariant, then μ is a convex combination of the Haar measure and an atomic probability measure supported by rational points. The same conclusion holds under the weaker assumption that μ is ν-stationary, i.e. μ=νμ, where ν is a finitely supported, probability measure on Γ whose support suppν generates Γ. The approach works more generally for Γ<SLd(Z).

Soit Γ un sous-groupe non-élementaire du groupe SL2(Z). Soit μ une mesure de probabilité Γ-invariante sur le tore T2. On démontre que μ est une moyenne de la mesure de Haar et une probabilité discrète portée par des points rationnels. La même conclusion reste vraie sous l'hypothèse que μ est ν-stationnaire, donc μ=νμ, où ν est une probabilité sur Γ à support fini et engendrant Γ. L'approche se généralise aux sous-groupes Γ de SLd(Z).

Published online:
DOI: 10.1016/j.crma.2007.04.017
Jean Bourgain 1; Alex Furman 2; Elon Lindenstrauss 3; Shahar Mozes 4

1 Institute for Advanced Study, Princeton, NJ 08540, USA
2 University of Illinois at Chicago, Chicago, IL 60607, USA
3 Princeton University, Princeton, NJ 08544, USA
4 The Hebrew University, 91904 Jerusalem, Israel
     author = {Jean Bourgain and Alex Furman and Elon Lindenstrauss and Shahar Mozes},
     title = {Invariant measures and stiffness for {non-Abelian} groups of toral automorphisms},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {737--742},
     publisher = {Elsevier},
     volume = {344},
     number = {12},
     year = {2007},
     doi = {10.1016/j.crma.2007.04.017},
     language = {en},
AU  - Jean Bourgain
AU  - Alex Furman
AU  - Elon Lindenstrauss
AU  - Shahar Mozes
TI  - Invariant measures and stiffness for non-Abelian groups of toral automorphisms
JO  - Comptes Rendus. Mathématique
PY  - 2007
SP  - 737
EP  - 742
VL  - 344
IS  - 12
PB  - Elsevier
DO  - 10.1016/j.crma.2007.04.017
LA  - en
ID  - CRMATH_2007__344_12_737_0
ER  - 
%0 Journal Article
%A Jean Bourgain
%A Alex Furman
%A Elon Lindenstrauss
%A Shahar Mozes
%T Invariant measures and stiffness for non-Abelian groups of toral automorphisms
%J Comptes Rendus. Mathématique
%D 2007
%P 737-742
%V 344
%N 12
%I Elsevier
%R 10.1016/j.crma.2007.04.017
%G en
%F CRMATH_2007__344_12_737_0
Jean Bourgain; Alex Furman; Elon Lindenstrauss; Shahar Mozes. Invariant measures and stiffness for non-Abelian groups of toral automorphisms. Comptes Rendus. Mathématique, Volume 344 (2007) no. 12, pp. 737-742. doi : 10.1016/j.crma.2007.04.017.

[1] D. Berend Multi-invariant sets on tori, Trans. Amer. Math. Soc., Volume 280 (2000) no. 2, pp. 509-532

[2] P. Bougerol; J. Lacroix Products of Random Matrices with Applications to Schrödinger Operators, Birkhäuser, 1985

[3] J. Bourgain On the Erdös–Volkmann and Katz–Tao ring conjecture, GAFA, Volume 13 (2003), pp. 334-365

[4] J. Bourgain, A. Gamburd, On the spectral gap for finitely-generated subgroups of SU(2), preprint, Invent., submitted for publication

[5] M. Einsiedler; E. Lindenstrauss Rigidity properties of Zd-actions on tori and solenoids, Electron. Res. Announc. Amer. Math. Soc., Volume 9 (2003), pp. 99-110

[6] H. Furstenberg Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, Volume 1 (1967), pp. 1-49

[7] H. Furstenberg Stiffness of group actions, Mumbai, 1996 (Tata Inst. Fund. Res. Stud. Math.), Volume vol. 14, Tata Inst. Fund. Res., Bombay (1998), pp. 105-117

[8] I.Ya. Gol'dsheĭd; G.A. Margulis Lyapunov exponents of a product of random matrices, Uspekhi Mat. Nauk, Volume 44 (1989) no. 5(269), pp. 13-60 (in Russian). Translation in Russian Math. Surveys, 44, 5, 1989, pp. 11-71

[9] Y. Guivarc'h, private communication

[10] Y. Guivarc'h; A.N. Starkov Orbits of linear group actions, random walk on homogeneous spaces, and toral automorphisms, Ergodic Theory Dynam. Systems, Volume 24 (2004) no. 3, pp. 767-802

[11] B. Kalinin; A. Katok Invariant measures for actions of higher rank abelian groups, Seattle, WA, 1999 (Proc. Sympos. Pure Math.), Volume vol. 69, Amer. Math. Soc. Providence, RI (2001), pp. 593-637

[12] A. Katok; R. Spatzier Invariant measures for higher rank hyperbolic abelian actions, Ergodic Theory Dynam. Systems, Volume 16 (1996) no. 4, pp. 751-778

[13] N. Katz; T. Tao Some connections between Falconer's distance set conjecture and sets of Furstenberg type, New York J. Math., Volume 7 (2001), pp. 149-187

[14] G.A. Margulis Problems and conjectures in rigidity theory, Mathematics: Frontiers and Perspectives, Amer. Math. Soc., 2000, pp. 161-174

[15] R. Muchnik, Orbits of Zariski dense semigroups of SL(n,Z), Ergodic Theory Dynam. Systems

[16] R. Muchnik Semigroup actions on Tn, Geom. Dedicata, Volume 110 (2005), pp. 1-47

[17] R. Pink Strong approximation for Zariski dense subgroups over arbitrary global fields, Comment. Math. Helv., Volume 75 (2000) no. 4, pp. 608-643

[18] D. Rudolph ×2 and ×3 invariant measures and entropy, Ergodic Theory Dynam. Systems, Volume 10 (1990) no. 2, pp. 395-406

[19] A.N. Starkov, Orbit closures of toral automorphism groups, preprint, Moscow, 1999

[20] B. Weisfeiler Strong approximation for Zariski-dense subgroups of semisimple algebraic groups, Ann. of Math. (2), Volume 120 (1984) no. 2, pp. 271-315

Cited by Sources:

This research is supported in part by NSF DMS grants 0627882 (JB), 0604611 (AF), 0500205 & 0554345 (EL) and BSF grant 2004-010 (SM).

Comments - Policy

Articles of potential interest

Mesures stationnaires et fermés invariants des espaces homogènes

Yves Benoist; Jean-François Quint

C. R. Math (2009)

Mesures stationnaires et fermés invariants des espaces homogènes II

Yves Benoist; Jean-François Quint

C. R. Math (2011)