Multi-invariant sets on tori

Given a compact metric group G, we are interested in those semigroups L of continuous endomorphisms of G, possessing the following property: The only infinite, closed, :-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.