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 $h_H\ge 1$ be an ample Weil height. Let ${\lambda }_1(f)$ denote the first dynamical degree of $f$. We show that there is a constant $C_{f,h_H}>0$ such that, for any $n\ge 1$ and any point $P\in X(\overline{K})$ with well-defined forward orbit, $h_H(f^n(P))\le C_{f,h_H}{h}_H(P)n^{2(d-1)}{\lambda }_1(f{)}^n$ if ${\lambda }_1(f)>1$, and $h_H(f^n(P))\le C_{f,h_H}{h}_H(P)n^{2d}$ if ${\lambda }_1(f)=1$. 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.

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