Density of orbits of dominant regular self-maps of semiabelian varieties

Abstract

We prove a conjecture of Medvedev and Scanlon in the case of regular morphisms of semiabelian varieties. That is, if $G$ is a semiabelian variety defined over an algebraically closed field $K$ of characteristic $0$, and $φ\colon G\to G$ is a dominant regular self-map of $G$ which is not necessarily a group homomorphism, we prove that one of the following holds: either there exists a non-constant rational fibration preserved by $φ$, or there exists a point $x\in G(K)$ whose $φ$-orbit is Zariski dense in $G$.