# Must the coordinates of a polynomial iteration have about the same size?

Original post. The following statements seem plausible (not to say intuitively obvious), but I do not see how to prove them.

Let us say that a polynomial mapping of $\mathbb{C}^2$ is reducible if either it takes the shape $(x,y) \mapsto (f(x),B(x,y))$ or the shape $(x,y) \mapsto (A(x,y),g(y))$, where $f,g$ are univariate polynomials.

(a) Consider $F : \mathbb{C}^2 \to \mathbb{C}^2$ a polynomial map. Write $F^{\circ N} = (A_N,B_N)$ coordinate-wise. If $F$ is not reducible, must $$\frac{\log{\deg{A_N}}}{\log{\deg{B_N}}} \to 1?$$ (b) Assume now $F$ is defined over the algebraic numbers $\bar{\mathbb{Q}}$, and let $h$ denote the absolute logarithmic Weil height. Consider $P \in \mathbb{A}^2(\bar{\mathbb{Q}})$ having Zariski-dense orbit under $F$. If $F$ is not reducible, must $$\frac{\log{h(A_N(P))}}{\log{h(B_N(P))}} \to 1?$$

Added. The following remark suggest that (b) might be non-trivial even for morphisms of $\mathbb{P}^2$ or $\mathbb{P}^1 \times \mathbb{P}^1$. Here is a counterpart of question (b) for abelian varieties which, to my knowledge, has not been given attention in the published literature.

Let $A/\bar{\mathbb{Q}}$ be an abelian variety and $\lambda : A \dashrightarrow \mathbb{A}^1$ a non-constant rational function (a coordinate). Suppose $P \in A(\bar{\mathbb{Q}})$ is such that $h(\lambda([n_i]P)) = o(n_i^2)$ along some sequence of integers $n_i$ with $|n_i| \to \infty$. Must there be a homomorphism $A \twoheadrightarrow B$ onto a positive-dimensional abelian variety mapping $P$ to a torsion point?

This contains the Mordell-Lang problem (proved by Faltings and Vojta) for the group $\Gamma = \langle P \rangle$ and the hypersurfaces of the form $\lambda^{-1}(\mathrm{pt})$. So it is certainly a deep statement.