Height growth of rational self-maps

Abstract

Let $K$ be a number field or a function field of a curve. Let $f$ be a dominant rational self-map of a normal projective variety $X$ of dimension $d$ over $K$, and let $h_H\geq1$ be an ample Weil height. Let $λ_1(f)$ denote the first dynamical degree of $f$. We show that there is a constant $C_{f,h_H}>0$ such that, for any $n\geq1$ and any point $P\in X(\overline{K})$ with well-defined forward orbit, $h_H(f^n(P))\leq C_{f,h_H}h_H(P)n^{2(d-1)}λ_1(f)^n$ if $λ_1(f)>1$, and $h_H(f^n(P))\leq C_{f,h_H}h_H(P)n^{2d}$ if $λ_1(f)=1$. Over number fields, the proof combines a new uniform upper bound for arithmetic degrees with a uniform comparison of point heights. Over function fields, it relies on the corresponding uniform upper bound for geometric degrees on a projective model.