  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.
