Silverman's conjecture for additive polynomial mappings

Abstract

Let $F : \mathrm{End}_{\mathbb{F_p}}(\mathbb{G}_{a/K}^d)$ be an additive polynomial mapping over a global function field $K/\mathbb{F}_q$, and let $P \in \mathbb{G}_a^d(K)$. Following Silverman, consider $\delta := \lim_{n \in \mathbb{N}} (\deg{F^{n}})^{1/n}$ the dynamic degree of $F$ and $\alpha(P) := \limsup_{n \in \mathbb{N}} h_K(F^{n}P)^{1/n}$ the arithmetic degree of $F$ at $P$. We have $\alpha(P) \leq \delta$, and extending a conjecture of Silverman from the number field case, it is expected that equality holds if the orbit of $P$ is Zariski-dense. We prove a weaker form of this conjecture: if $\delta > 1$ and the orbit of $P$ is Zariski-dense, then also $\alpha(P) > 1$. We obtain furthermore a more precise result concerning the growth along the orbit of $P$ of the heights of the individual coordinates, and formulate a few related open problems motivated by our results, including a generalization "with moving targets" of Faltings's theorem back in the number field case.