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
$$
\frac{\log{h(A_N(P))}}{\log{h(B_N(P))}} \to 1?
$$