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

Published online by Cambridge University Press:  20 September 2007

Rafe Jones
Affiliation:
Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI 53706, USA [email protected]
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Abstract

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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.

Type
Research Article
Copyright
Foundation Compositio Mathematica 2007