Invariant measures and stiffness for non-Abelian groups of toral automorphisms

Jean Bourgain; Alex Furman; Elon Lindenstrauss; Shahar Mozes

doi:10.1016/j.crma.2007.04.017

Group Theory

Invariant measures and stiffness for non-Abelian groups of toral automorphisms

Presented by: Jean Bourgain

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

Comptes Rendus. Mathématique, Volume 344 (2007) no. 12, pp. 737-742.

Abstract
Résumé

Let Γ be a non-elementary subgroup of ${SL}_{2} (Z)$ . If $μ$ is a probability measure on $T^{2}$ 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. $μ = ν * μ$ , where $ν$ is a finitely supported, probability measure on Γ whose support suppν generates Γ. The approach works more generally for $Γ < {SL}_{d} (Z)$ .

Soit Γ un sous-groupe non-élementaire du groupe ${SL}_{2} (Z)$ . Soit $μ$ une mesure de probabilité Γ-invariante sur le tore $T^{2}$ . 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 $μ = ν * μ$ , où $ν$ est une probabilité sur Γ à support fini et engendrant Γ. L'approche se généralise aux sous-groupes Γ de ${SL}_{d} (Z)$ .

Received: 2007-04-11
Accepted: 2007-04-24
Published online: 2007-06-15

DOI: 10.1016/j.crma.2007.04.017

Author's affiliations:

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

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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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.04.017/

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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).

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