Multi-Invariant Sets on Tori

Given a compact metric group $G$, we are interested in those semigroups $\Sigma$ of continuous endomorphisms of $G$, possessing the following property: The only infinite, closed, $\Sigma$-invariant subset of $G$ is $G$ itself. Generalizing a one-dimensional result of Furstenberg, we give here a full characterization—for the case of finitedimensional tori—of those commutative semigroups with the aforementioned property.