Density of orbits of endomorphisms of abelian varieties (Q2826766)

Let \(K_0\) be an algebraically closed field of characteristic zero. Let \(A\) be an abelian variety defined over \(K_0\), and let \(\sigma:A\rightarrow A\) be a dominant map. For any \(x\in A(K_0)\), let \(\mathcal{O}_{\sigma}(x)\) denote the forward \(\sigma\)-orbit, namely the set of all \(\sigma^n(x)\) for \(n\geq 0\).NEWLINENEWLINEThe main result of this paper shows that the following statements are equivalent: (1) there exists \(x\in A(K_0)\) such that \(\mathcal{O}_{\sigma}(x)\) is Zariski dense in \(A\). (2) there exists no non-constant rational map \(f:A\rightarrow\mathbb{P}^1\) such that \(f\circ\sigma = f\). This provides a positive answer for abelian varieties of a question raised by \textit{A. Medvedev} and the second author in [Ann. Math. (2) 179, No. 1, 81--177 (2014; Zbl 1347.37145)]. The authors prove also a stronger statement of this result in which \(\sigma\) is replaced by any commutative finitely generated monoid of dominant endomorphisms of \(A\).