Iterated Galois towers, their associated martingales, and the p-adic Mandelbrot set

We study the Galois tower generated by iterates of a quadratic polynomial f defined over an arbitrary field. One question of interest is to find the proportion an of elements at level n that fix at least one root; in the global field case these correspond to unramified primes in the base field that have a divisor at level n of residue class degree one. We thus define a stochastic process associated to the tower that encodes root-fixing information at each level. We develop a uniqueness result for certain permutation groups, and use this to show that for many f each level of the tower contains a certain central involution. It follows that the associated stochastic process is a martingale, and convergence theorems then allow us to establish a criterion for showing that an tends to 0. As an application, we study the dynamics of the family x2 + c ∈ Fp[x], and this in turn is used to establish a basic property of the p-adic Mandelbrot set.