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

Abstract

Let ν \nu be a probability measure on S L d ( Z ) \mathrm {SL}_d(\mathbb {Z}) satisfying the moment condition E ν ( ‖ g ‖ ϵ ) > ∞ \mathbb {E}_\nu (\|g\|^\epsilon )>\infty for some ϵ \epsilon . We show that if the group generated by the support of ν \nu is large enough, in particular if this group is Zariski dense in S L d \mathrm {SL}_d , for any irrational x ∈ T d x \in \mathbb {T}^d the probability measures ν ∗ n ∗ δ x \nu ^{* n} * \delta _x tend to the uniform measure on T d \mathbb {T}^d . If in addition x x is Diophantine generic, we show this convergence is exponentially fast.