Propagation of Zariski Dense Orbits

TL;DR

This paper investigates the propagation of Zariski dense orbits under morphisms on smooth projective varieties over number fields, establishing results that many points have dense orbits if one does, for various classes of varieties.

Contribution

It formulates and proves new results on the density propagation of orbits for different classes of varieties and morphisms, extending understanding of dynamical systems over number fields.

Findings

If one point has a Zariski dense orbit, many do as well.

Results hold for projective spaces, abelian varieties, and surfaces.

Sets of representatives for dense orbits can be Zariski dense.

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 $\mathcal{O}_f(P_0):=\bigl\{f^n(P):n\in\mathbb{N}\bigr\}$ is Zariski dense, then there are many such points. 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 is Zariski dense. We prove statements of this sort for various classes of varieties and maps, including projective spaces, abelian varieties, and surfaces.