# Density of orbits of endomorphisms of commutative linear algebraic groups

## TL;DR

This paper proves a conjecture for endomorphisms of connected commutative linear algebraic groups over algebraically closed fields of characteristic zero, showing that either a rational function is preserved or a dense orbit exists.

## Contribution

It establishes the dichotomy for dominant endomorphisms of such groups, confirming the Medvedev-Scanlon conjecture in this setting.

## Key findings

- Either a non-constant rational function is invariant under the endomorphism.
- Or there exists a point with a Zariski dense orbit.
- The result applies to all connected commutative linear algebraic groups over algebraically closed fields of characteristic zero.

## Abstract

We prove a conjecture of Medvedev and Scanlon for endomorphisms of connected commutative linear algebraic groups $G$ defined over an algebraically closed field $\mathbb{k}$ of characteristic $0$. That is, if $\Phi\colon G\longrightarrow G$ is a dominant endomorphism, we prove that one of the following holds: either there exists a non-constant rational function $f\in \mathbb{k}(G)$ preserved by $\Phi$ (i.e., $f\circ \Phi = f$), or there exists a point $x\in G(\mathbb{k})$ whose $\Phi$-orbit is Zariski dense in $G$.