Density of orbits of endomorphisms of abelian varieties

TL;DR

This paper proves that for a dominant endomorphism of an abelian variety over algebraic numbers, either there is a preserved rational fibration or a point with a Zariski dense orbit, answering a previously posed question.

Contribution

It establishes a dichotomy for endomorphisms of abelian varieties, confirming a conjecture and extending results to finitely generated monoids of endomorphisms.

Findings

Existence of a dense orbit or preserved fibration for endomorphisms

Extension of results to finitely generated monoids

Positive answer to a question by Medvedev and Zhang

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 ("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 $A$.