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

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

Jean Bourgain^*, Alex Furman, Elon Lindenstrauss, Shahar Mozes

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

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 = 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). To cite this article: J. Bourgain et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Original language	American English
Pages (from-to)	737-742
Number of pages	6
Journal	Comptes Rendus Mathematique
Volume	344
Issue number	12
DOIs	https://doi.org/10.1016/j.crma.2007.04.017
State	Published - 15 Jun 2007

Bibliographical note

Funding Information:
✩ 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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10.1016/j.crma.2007.04.017

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title = "Invariant measures and stiffness for non-Abelian groups of toral automorphisms",

abstract = "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 = ν * μ, where ν is a finitely supported, probability measure on Γ whose support suppν generates Γ. The approach works more generally for Γ < SLd (Z). To cite this article: J. Bourgain et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).",

author = "Jean Bourgain and Alex Furman and Elon Lindenstrauss and Shahar Mozes",

note = "Funding Information: ✩ 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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T1 - Invariant measures and stiffness for non-Abelian groups of toral automorphisms

AU - Bourgain, Jean

AU - Furman, Alex

AU - Lindenstrauss, Elon

AU - Mozes, Shahar

N1 - Funding Information: ✩ This research is supported in part by NSF DMS grants 0627882 (JB), 0604611 (AF), 0500205 & 0554345 (EL) and BSF grant 2004-010 (SM).

PY - 2007/6/15

Y1 - 2007/6/15

N2 - 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 = ν * μ, where ν is a finitely supported, probability measure on Γ whose support suppν generates Γ. The approach works more generally for Γ < SLd (Z). To cite this article: J. Bourgain et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).

AB - 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 = ν * μ, where ν is a finitely supported, probability measure on Γ whose support suppν generates Γ. The approach works more generally for Γ < SLd (Z). To cite this article: J. Bourgain et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).

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ER -

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

Abstract

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