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Comptes Rendus Mathématique
Volume 344, n° 12
pages 737-742 (juin 2007)
Doi : 10.1016/j.crma.2007.04.017
Received : 11 April 2007 ;  accepted : 24 April 2007
Invariant measures and stiffness for non-Abelian groups of toral automorphisms
Mesures invariantes et rigidité pour groupes non-abeliens dʼautomorphismes du tore
 

Jean Bourgain a, Alex Furman b, Elon Lindenstrauss c, Shahar Mozes d
a Institute for Advanced Study, Princeton, NJ 08540, USA 
b University of Illinois at Chicago, Chicago, IL 60607, USA 
c Princeton University, Princeton, NJ 08544, USA 
d The Hebrew University, 91904 Jerusalem, Israel 

Abstract

Let Γ be a non-elementary subgroup of  . If μ is a probability measure on   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 -stationary, i.e.  , where is a finitely supported, probability measure on Γ whose support supp generates Γ . The approach works more generally for  . To cite this article: J. Bourgain et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).

The full text of this article is available in PDF format.
Résumé

Soit Γ un sous-groupe non-élementaire du groupe  . Soit μ une mesure de probabilité Γ -invariante sur le tore  . 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 lʼhypothèse que μ est -stationnaire, donc  , où est une probabilité sur Γ à support fini et engendrant Γ . Lʼapproche se généralise aux sous-groupes Γ de  . Pour citer cet article : J. Bourgain et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).

The full text of this article is available in PDF format.

 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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