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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus
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by Jean Bourgain, Alex Furman, Elon Lindenstrauss and Shahar Mozes
J. Amer. Math. Soc. 24 (2011), 231-280
DOI: https://doi.org/10.1090/S0894-0347-2010-00674-1
Published electronically: June 29, 2010

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.
References
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Bibliographic 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
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2726604