Multi-Invariant Sets on Compact Abelian Groups

Let $G$ be a finite-dimensional connected compact abelian group. Generalizing previous results, dealing with the case of finite-dimensional tori, a full characterization is given herewith of those commutative semigroups $\Sigma$ of continuous endomorphisms of $G$ which satisfy the following property: The only infinite closed $\Sigma$-invariant subset of $G$ is $G$ itself.