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

TL;DR

This paper proves a conjecture regarding the behavior of dominant regular self-maps on semiabelian varieties, showing conditions under which orbits are dense or preserved fibrations exist.

Contribution

It establishes the conjecture of Medvedev and Scanlon for regular morphisms of semiabelian varieties, extending understanding of orbit density and fibrations.

Findings

Either a non-constant rational fibration is preserved by the map

Or there exists a point with a Zariski dense orbit

The result applies over algebraically closed fields of characteristic zero

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 $\varphi\colon G\to G$ 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 $\varphi$, or there exists a point $x\in G(K)$ whose $\varphi$-orbit is Zariski dense in $G$.