Semigroup actions on tori and stationary measures on projective spaces

Let Gamma be a subsernigroup of G = GL(d, R), d > 1. We assume that the action of Gamma on R-d is strongly irreducible and that Gamma contains a proximal and quasi-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 p(d-1). In the case where Gamma is a subsemigroup of GL(d,R) boolean AND M(d,Z) and Gamma has the above properties, we deduce that the Gamma-orbits on T-d = R-d/Z(d) are finite or dense.