Galois martingales and the hyperbolic subset of the p-adic Mandelbrot set

The study of iteration of rational functions with complex coefficients has garnered much attention in the past twenty-five years. More recently, a parallel theory over the p-adics has emerged, bearing interesting similarities and dissimilarities to the complex theory. One of the best-known objects in complex dynamics is the Mandelbrot set, defined to be the collection of complex numbers c such that 0 has a bounded orbit under iteration of x2 + c. Its hyperbolic subset consists of c such that 0 tends to an attracting cycle. This much-studied subset occupies most of the area of the Mandelbrot set. In this thesis I examine the p-adic analogue of the hyperbolic subset of the Mandelbrot set, and I show that it has density zero in a natural sense, putting it in sharp contrast to the complex case.;The method of proof appears to be highly unusual. First, elementary arguments involving the reduction homomorphism reduce the problem to studying the dynamics of x2 + c where c is an element of the algebraic closure of the finite field with p elements. The Tchebotarev Density theorem gives a further translation of the problem into one involving a certain tower of Galois extensions of function fields. I then define a stochastic process associated to this Galois tower. I show that this stochastic process is a martingale, then use a martingale convergence theorem to obtain the desired result.