The height of an orbit under rational self-maps

Question

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.

Answer 1

I thank Mahdi for the pointer to the paper. It was my first paper on this subject. It considers especially the case of monomial maps. It contains some references to a small number of papers by other people who have studied the growth rate of $h(\phi^n(x_0))$.

Dynamical Degrees, Arithmetic Degrees, and Canonical Heights for Dominant Rational Self-Maps of Projective Space, http://arxiv.org/abs/1111.5664

In the general setting $\phi:X\to X$, let $x_0$ be a point whose entire forward orbit is well-defined. A somewhat coarse, but still quite interesting, measure of the growth rate of $h(\phi^n(x_0))$ is the arithmetic degree, which by definition is the limit (if the limit exists) $$ \alpha_\phi(x_0) = \lim_{n\to\infty} h(\phi^n(x_0))^{1/n}. $$ Shu Kawaguchi and I have studied the arithmetic degree, and its relation to the geometrically defined dynamical degree $\delta_\phi$ of $\phi$, in several papers. For example, we proved in general that $\alpha_\phi(x_0)\le\delta_\phi$ (this is easy for projective space, but gets harder if the Neron-Severi group has rank larger than 1), we proved that for morphisms, the limit defining $\alpha_\phi(x_0)$ always exists and is an algebraic integer, and we proved that if $X=E^N$ is a power of a non-CM elliptic curve and $\phi$ is an isogeny and the orbit of $x_0$ is Zariski dense, then there is equality $\alpha_\phi(x_0)=\delta_\phi$. (We conjecture these properties are true in general.)

Here are the links. All are joint with Shu Kawaguchi.

On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties, http://arxiv.org/abs/1208.0815

Examples of dynamical degree equals arithmetic degree, http://arxiv.org/abs/1212.3015

Dynamical Canonical Heights for Jordan Blocks and Arithmetic Degrees of Orbits, http://arxiv.org/abs/1301.4964

Regarding the statement in your edit, are you using anything more than the fact that the orbit is contained in $X(K)$ for some number field $K$, and then apply the trivial upper bound for the number of $K$-rational points in projective space? I think that statements of this sort can be interesting, but probably the $o(\log n)$ needs to be replaced by a function that grows significantly faster to get a really interesting statement.

Answer 2

Thank you very much for this answer!

This is not going to fit in a comment, so I'm writing another answer.

[EDIT: I had hastily written some incorrect statements, now corrected in the text below. My apology for those mistakes. ]

1. I see from your conjecture that as soon as the dynamical degree $\delta_{\varphi} > 1$ and the orbit is Zariski dense, the height - conjecturally - ought to grow exponentially. On the other hand, when $\delta_{\varphi} = 1$, one needs a finer measure for the growth of both $\deg{\varphi^{n}}$ and $h(\varphi^n(x_0))$. Regarding the degree, we expect that as soon as $\deg{\varphi^n}$ is unbounded (which I guess will be the case as soon as $\varphi$ is not an birational?), it should grow at least linearly in $n$. But can one even show, unconditionally, that there exists an explicit function $\lambda(n)$, going monotonously to $\infty$, such that, for every $(X,\varphi)$ with $\deg{\varphi^n}$ unbounded, it holds $\deg{\varphi^n} > \lambda(n)$ for infinitely many $n$? I realize I can show this, say (for concreteness) with $\lambda(n) = \log{n}$.

2. Thus, my question mostly concerned the case $\delta_{\varphi} = 1$. It is of course possible, in that case, to have a Zariski-dense orbit with height growing logarithmically. (But I can prove in such a case, say for $X = \mathbb{P}^r$, if $h(\varphi^n(x_0)) = O(\log{n})$, then upon restricting $n$ to an arithmetic progression, the coordinates of $\varphi^n(x_0)$ are polynomials in $n$. In general, $\varphi$ must be birational. So this case is very special.)

3. Regarding the edit. I can actually show this with the $o(\log{n})$ term replaced by $o(n^{1/\mathrm{dim}(X)})$ --- but then the conclusion must be modified to include examples like translations of the affine line or automorphisms of $\mathbb{A}^2$ such as $(x,y) \mapsto (x+1,y+x^2)$. In effect: if $\varphi : X \dashrightarrow X$ is a rational self-map such that the orbit of $x_0$ is Zariski dense and contained in the domain of $\varphi$, and if $h(\varphi^n(x_0)) = o(n^{1/\dim(X)})$, then $\varphi$ is birational. (I wrote $o(\log{n})$ in my question only to retain the stronger conclusion that forces $x_0$ to be pre-periodic). To be sure, the same conclusion should be expected under the weaker bound $h(\varphi^n(x_0)) = o(n)$ --- but I can't prove this.

4. In particular: $\lambda(n) = \log{n}$ is the optimal function such that, for an arbitrary $(X,\varphi)$, the pre-periodic points can be characterized by $\lambda(n) - h(\varphi^n(x_0)) \to +\infty$.