A lower bound on the orbit growth of a regular self-map of affine space

Abstract

We show that if $f : \mathbb{A}_{\bar{\mathbb{Q}}}^r \to \mathbb{A}_{\bar{\mathbb{Q}}}^r$ is a regular self-map and $P \in \mathbb{A}^r(\bar{\mathbb{Q}})$ has $\limsup_{n \in \mathbb{N}} \frac{\log{h_{\mathrm{aff}}(f^nP)}}{\log{n}} < 1/r$, where $h_{\textrm{aff}}$ is the affine Weil height, then $\mathbb{N}$ partitions into a finite set and finitely many full arithmetic progressions, on each of which the coordinates of $f^nP$ are polynomials in $n$. In particular, if $(f^nP)_{n \in \mathbb{N}}$ is a Zariski-dense orbit, then either $n = 1$ and $f$ is of the shape $t \mapsto \zeta t + c$, $\zeta \in \mu_{\infty}$, or else $\limsup_{n \in \mathbb{N}} \frac{\log{h_{\mathrm{aff}}(f^nP)}}{\log{n}} \geq 1/r$. This inequality is the exponential improvement of the trivial lower bound obtained from counting the points of bounded height in $\mathbb{A}^r(K)$.