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).

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Published online:
DOI: 10.1016/j.crma.2007.04.017
Bourgain, Jean 1; Furman, Alex 2; Lindenstrauss, Elon 3; Mozes, Shahar 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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Bourgain, Jean; Furman, Alex; Lindenstrauss, Elon; Mozes, Shahar. 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. http://www.numdam.org/articles/10.1016/j.crma.2007.04.017/

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