Semigroup actions on tori and stationary measures on projective spaces

Abstract

Let $\Gamma$ be a sub-semigroup of $G=GL(d,\mathbb R),$ $d>1.$ We assume that the action of $\Gamma$ on $\R^d$ is strongly irreducible and that $\Gamma$ contains a proximal and expanding element. We describe contraction properties of the dynamics of $\Gamma$ on $\R^d$ at infinity. This amounts to the consideration of the action of $\Gamma$ on some compact homogeneous spaces of $G,$ which are extensions of the projective space $\pr^{d-1}.$ In the case where $\Gamma$ is a sub-semigroup of $GL(d,\R)\cap M(d,\Z)$ and $\Gamma$ has the above properties, we deduce that the $\Gamma$-orbits on $\T^d=\R^d\slash\Z^d$ are finite or dense.