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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(\Vert g\Vert^\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

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

Shahar Mozes
Affiliation: Department of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel

DOI: https://doi.org/10.1090/S0894-0347-2010-00674-1
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.

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