Existence of non-preperiodic algebraic points for a rational self-map of infinite order

TL;DR

The paper proves that for a dominant rational self-map of infinite order on a variety over a number field, there are many algebraic points that are not preperiodic, using an elementary proof approach.

Contribution

It introduces an elementary proof demonstrating the existence of non-preperiodic algebraic points for rational self-maps of infinite order, extending beyond the regular polarized case.

Findings

Existence of many non-preperiodic algebraic points under such maps

Elementary proof method independent of canonical height theory

Applicable to general rational self-maps, not just regular polarized ones

Abstract

Let $X$ be a variety defined over a number field and $f$ be a dominant rational self-map of $X$ of infinite order. We show that $X$ admits many algebraic points which are not preperiodic under $f$. If $f$ were regular and polarized, this would follow immediately from the theory of canonical heights, but it does not work very well for rational self-maps. We provide an elementary proof following an argument by Bell, Ghioca and Tucker (arxiv:0808.3266).