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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.

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Have you seen this? It doesn't say much about the affine case, but it might be interesting if you haven't seen it already. –  Mahdi Majidi-Zolbanin Jan 30 '13 at 16:47
Thanks for the reference - I hadn't seen it! I see now that I should frame the question, more generally, to address rational self-maps $\varphi : X \dashrightarrow X$ such that the orbit of $x_0$ lies in the domain of $\varphi$. –  Vesselin Dimitrov Jan 30 '13 at 17:08

2 Answers 2

up vote 9 down vote accepted

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,

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,

Examples of dynamical degree equals arithmetic degree,

Dynamical Canonical Heights for Jordan Blocks and Arithmetic Degrees of Orbits,

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.

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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 $\lambda(n) - h(\varphi^n(x_0)) \to +\infty$.

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