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

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).