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