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Iterated Galois towers, their associated martingales, and the \(p\)-adic Mandelbrot set. (English) Zbl 1166.11040
Summary: 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 \(a_n\) 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 \(a_n\) tends to 0. As an application, we study the dynamics of the family \(x^2 + c \in\overline{\mathbb{F}}_p[x]\), and this in turn is used to establish a basic property of the \(p\)-adic Mandelbrot set.

11R32 Galois theory
37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
60G42 Martingales with discrete parameter
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