Propagation of Zariski dense orbits

Abstract

Let $X/K$ be a smooth projective variety defined over a number field, and let $f:X\to X$ be a morphism defined over $K$. We formulate a number of statements of varying strengths asserting, roughly, that if there is at least one point $P_0\in X(K)$ whose $f$-orbit ${\mathcal{O}}_f(P_0):=\{f^n(P):n\in \mathbb{N}\}$ is Zariski dense, then after replacing $K$ by a finite extension, there are many $f$-orbits in $X(K)$. For example, a weak conclusion would be that $X(K)$ is not the union of finitely many (grand) $f$-orbits, while a strong conclusion would be that any set of representatives for the Zariski dense grand $f$-orbits in $X(K)$ is itself Zariski dense. We prove statements of this sort for various classes of varieties and maps, including projective spaces, abelian varieties, and surfaces.