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

Bourgain, Jean; Furman, Alex; Lindenstrauss, Elon; Mozes, Shahar

doi:10.1090/S0894-0347-2010-00674-1

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.

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