Density of orbits of endomorphisms of abelian varieties

Abstract:

Let $A$ be an abelian variety defined over $\bar {\mathbb {Q}}$, and let $\varphi$ be a dominant endomorphism of $A$ as an algebraic variety. We prove that either there exists a non-constant rational fibration preserved by $\varphi$ or there exists a point $x\in A(\bar {\mathbb {Q}})$ whose $\varphi$-orbit is Zariski dense in $A$. This provides a positive answer for abelian varieties of a question raised by Medvedev and the second author. We also prove a stronger statement of this result in which $\varphi$ is replaced by any commutative finitely generated monoid of dominant endomorphisms of $A$.