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

TL;DR

This paper investigates the intersection patterns of orbits of rational self-maps on semiabelian varieties with finitely generated subgroups, revealing structured sets and finiteness under regularity conditions.

Contribution

It establishes that the intersection set is a union of finitely many arithmetic progressions plus a zero-density set, and proves finiteness when the map is regular.

Findings

The intersection set is a union of finitely many arithmetic progressions and a zero-density set.

Under regularity of the map, the intersection set is finite.

Provides a structural understanding of orbit-subgroup intersections in semiabelian varieties.

Abstract

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