## 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
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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.## 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