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

Abstract:

Let $K$ be a global function field and let $\phi (x)\in K[x]$. 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 $\mathcal {Z}(\phi ,b)$ that depends only on $\phi$, the poles of the $b$, and $K$. Moreover, when we order $b\in \mathcal {O}_{K,S}$ by height, we show that $\mathcal {Z}(\phi ,b)$ is empty on average. As an application, we prove that the inverse limit of the Galois groups of iterates of $\phi (x)=x^d+f$ is a finite index subgroup of an iterated wreath product of cyclic groups. In particular, since our methods translate to rational function fields in characteristic zero, we establish the inverse Galois problem for these groups via specialization.