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

Abstract

Let $G$ be a semiabelian variety defined over an algebraically closed field $K$, endowed with a rational self-map $\hat{I}¦$. Let $\hat{I}\pm \in G(K)$ and let $\hat{I}“\subseteq G(K)$ be a finitely generated subgroup. We show that the set $\{n\in \mathbb{N}:\hat{I}{¦}^n(\hat{I}\pm )\in \hat{I}“\}$ is a union of finitely many arithmetic progressions along with a set of Banach density equal to $0$. In addition, assuming $\hat{I}¦$ is regular, we prove that the set $S$ must be finite.

12 pages