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

Bourgain, Jean¹ ; Furman, Alex² ; Lindenstrauss, Elon³ ; Mozes, Shahar⁴

¹ School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
² Department of Mathematics, University of Illinois at Chicago, 51 S Morgan Street, MSCS (m/c 249), Illinois 60607
³ Department of Mathematics, Princeton University, Princeton, New Jersey 08544, and Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel
⁴ Department of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel

Journal of the American Mathematical Society, Tome 24 (2011) no. 1, pp. 231-280

Résumé

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.

Détail

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

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