  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).
