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Journal of the American Mathematical Society

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Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus


Authors: Jean Bourgain, Alex Furman, Elon Lindenstrauss and Shahar Mozes
Journal: J. Amer. Math. Soc. 24 (2011), 231-280
MSC (2010): Primary 11B75, 37A17; Secondary 37A45, 11L07, 20G30
DOI: https://doi.org/10.1090/S0894-0347-2010-00674-1
Published electronically: June 29, 2010
MathSciNet review: 2726604
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Abstract: Let $\nu$ be a probability measure on $\mathrm {SL}_d(\mathbb {Z})$ satisfying the moment condition $\mathbb {E}_\nu (\|g\|^\epsilon )<\infty$ for some $\epsilon$. We show that if the group generated by the support of $\nu$ is large enough, in particular if this group is Zariski dense in $\mathrm {SL}_d$, for any irrational $x \in \mathbb {T}^d$ the probability measures $\nu ^{* n} * \delta _x$ tend to the uniform measure on $\mathbb {T}^d$. If in addition $x$ is Diophantine generic, we show this convergence is exponentially fast.


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Additional Information

Jean Bourgain
Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
MR Author ID: 40280

Alex Furman
Affiliation: Department of Mathematics, University of Illinois at Chicago, 51 S Morgan Street, MSCS (m/c 249), Illinois 60607

Elon Lindenstrauss
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544, and Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel
MR Author ID: 605709

Shahar Mozes
Affiliation: Department of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel
MR Author ID: 264125

Received by editor(s): November 18, 2009
Received by editor(s) in revised form: March 18, 2010
Published electronically: June 29, 2010
Additional Notes: The first author was supported in part by NSF grants DMS-0808042 and DMS-0835373
The second author was supported in part by NSF grants DMS-0604611 and DMS-0905977.
The third author was supported in part by NSF grants DMS-0554345 and DMS-0800345.
The fourth author was supported in part by BSF and ISF
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.