I have this basic question on which, strangely enough, the algebraic dynamics literature appears to be silent. But the question does not appear to be totally trivial or uninteresting to me - am I wrong?

*Question.* Let $\varphi : \mathbb{A}^r \to \mathbb{A}^r$ be a regular map defined over $\bar{\mathbb{Q}}$, and let $x_0 \in \mathbb{A}^r(\bar{{\mathbb{Q}}})$ be an algebraic point. What can be said about the growth, in $n$, of the (logarithmic) height of the iterates $h(\varphi^n(x_0))$? More generally: the same question with an endomorphism $\varphi : X \to X$ of any quasi-projective variety.

For example, one can obviously realize the growth rates $O(1)$ (iff the orbit is pre-periodic), $O(\log{n})$, $O(n)$, and $O(d^n)$ for all $d \in \mathbb{N}$.

The question is of course trivial for $r = 1$, or more generally if $\varphi$ extends to a morphism $\mathbb{P}^r \to \mathbb{P}^r$. Or more generally still, for $\varphi : X \to X$ with $X$ projective. (In this case, only the mentioned growth rates are possible).

One (e.g., I) can characterize the $\varphi$ with $h(\varphi^n(x_0))$ having a small growth rate (e.g., bounded by $O(n^{1/r})$), and I wondered whether this is of any interest, or completely trivial.

EDIT: More generally, consider rational self-maps $\varphi : X \dashrightarrow X$ of a projective variety $X$ over $\overline{\mathbb{Q}}$, and a point $x_0 \in X(\bar{\mathbb{Q}})$ whose orbit is contained in the domain of $\varphi$. Then I can show, for instance (is this self-evident?) that if $h(\varphi^n(x_0)) = o(\log{n})$, then $x_0$ is pre-periodic.

NEW EDIT (2/17): On returning to this question, I realized just now that the statement from the previous edit (from 1/30), as written, was indeed a trivial consequence of the rational point count and the pigeonhole principle, the latter forcing the characterization of pre-periodic points as above, with the $o(\log{n})$ improved by, roughly, $\frac{1}{\dim{X}}\log{n}$. Sorry about that. What I really wanted to say was not $o(\log{n})$, but (essentially) $\leq \log{n}$. In other words, the factor $1/\dim{X}$ in the trivial lower bound may be improved, in the setup of the previous edit, to $1$: more precisely, if $\log{n} - h(\varphi^n(x_0)) \to +\infty$, then $x_0$ is pre-periodic. It is this that I intended in my remark that the logarithm is the slowest growth rate of a non-preperiodic orbit. (Note that $h$ is the *logarithmic* height; thus, for a non-zero translation of $\mathbb{A}^1$, the height of the orbit is just $\log{n} + O(1)$.)

In fact, excluding certain basic, well understood cases, of which translations of $\mathbb{A}^1$ are the prototypical example, and in all of which the height is asymptotic to $d \log{n}$ for some $d \in \mathbb{N}$, the trivial lower bound $\log{(n^{1/\dim{X}})}$ can be improved *exponentially*, to $n^{1/\dim{X}}$.

Having realized that the statement in the previous edit was trivial (and uninteresting) as written, I just wanted to record those additional remarks here.