Periodic subvarieties of semiabelian varieties and annihilators of irreducible representations

Let $G$ be a semiabelian variety defined over a field of characteristic $0$, endowed with an endomorphism $\mathrm{\Phi }$. We prove there is no proper subvariety $Y\subset G$ which intersects the orbit of each periodic point of $G$ under the action of $\mathrm{\Phi }$. As an application, we are able to give a topological characterization of the annihilator ideals of irreducible representations in certain skew polynomial algebras.