Average Zsigmondy sets, dynamical Galois groups, and the Kodaira-Spencer map

Let $K$ be a global function field and let $ϕ\in K[x]$. For all wandering basepoints $b\in K$, we show that there is a bound on the size of the elements of the dynamical Zsigmondy set $\mathcal{Z}(ϕ,b)$ that depends only on $ϕ$, the poles of the $b$, and $K$. Moreover, when we order $b\in\mathcal{O}_{K,S}$ by height, we show that $\mathcal{Z}(ϕ,b)$ is empty on average. As an application, we prove that the inverse limit of the Galois groups of iterates of $ϕ(x)=x^d+f$ is a finite index subgroup of an iterated wreath product of cyclic groups. Finally, we establish similar results on Zsigmondy sets when $K$ is the field of rational numbers or $K$ is a quadratic imaginary field subject to an added stipulation: either zero has finite orbit under iteration of $ϕ$ or the Vojta conjecture for algebraic points on curves holds.

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