Orbits of linear group actions, random walks on homogeneous spaces and toral automorphisms

Let V be a finite-dimensional vector space over $\mathbb{R}$and let $\Gamma \subset \mathrm{GL}(V)$ be a semigroup. We study the closed $\Gamma$-invariant subsets of V-\{0\} under the condition that the Zariski closure of $\Gamma$ is semi-simple. We use the results to show that, if $\Gamma\subset\mathrm{SL}(\mathbb{R}^d)$ acts on $\mathbb{T}^d$ by automorphisms, then the orbits of $\Gamma$ are finite or dense.