Intersections of orbits of self-maps with subgroups in semiabelian varieties

Let G$G$ be a semiabelian variety defined over an algebraically closed field K$K$, endowed with a rational self-map phi$\Phi$. Let alpha is an element of G(K)$\alpha \in G(K)$ and let Gamma subset of G(K)$\Gamma \subseteq G(K)$ be a finitely generated subgroup. We show that the set {n is an element of N:phi n(alpha)is an element of Gamma}$\lbrace n\in \mathbb {N} \colon \Phi <^>n(\alpha)\in \Gamma \rbrace$ is a union of finitely many arithmetic progressions along with a set S$S$ of Banach density equal to 0$\hskip.001pt 0$. In addition, assuming that phi$\Phi$ is regular, we prove that the set S$S$ must be finite.