  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
("nvariant varieties for polynomial dynamical systems", Ann. of Math. (2) 179
(2014), no. 1, 81-177). We prove also a stronger statement of this result in
which $\varphi$ is replaced by any commutative finitely generated monoid of
dominant endomorphisms of $A$.
