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

We show that if $f : $\mathbb{A}${\bar{$\mathbb{Q}$}}^r \to $\mathbb{A}${\bar{$\mathbb{Q}$}}^r$isaregularself-mapand$P \in $\mathbb{A}$^r(\bar{$\mathbb{Q}$})$has$\limsup_{n \in $\mathbb{N}$} \frac{\log{h_{$\mathrm{a}\mathrm{f}\mathrm{f}$}(f^nP)}}{\log{n}} < 1/r$,where$h_{$\text{aff}$}$istheaffineWeilheight,then$$\mathbb{N}$$ partitions into a finite set and finitely many full arithmetic progressions, on each of which the coordinates of $f^{n}P$ are polynomials in $n$. In particular, if $(f^{n}P{)}_{n\in \mathbb{N}}$ is a Zariski-dense orbit, then either $n=1$ and $f$ is of the shape $t\mapsto \zeta t+c$, $\zeta \in \mu_{\infty}$,orelse$\limsup_{n \in $\mathbb{N}$} \frac{\log{h_{$\mathrm{a}\mathrm{f}\mathrm{f}$}(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 ${\mathbb{A}}^r(K)$.