Rafe Jones

The density of prime divisors in the arithmetic dynamics of quadratic polynomials

Let f [x], and consider the recurrence given by an = f(an � 1), with a0 . Denote by P(f, a0) the set of prime divisors of this recurrence, that is, the set of primes dividing at least one non-zero term, and denote the natural density of this set by D(P(f, a0)). The problem of determining D(P(f, a0)) when f is linear has attracted significant study, although it remains unresolved in full generality. In this paper, we consider the case of f quadratic, where previously D(P(f, a0)) was known only in a few cases. We show that D(P(f, a0)) = 0 regardless of a0 for four infinite families of f, including f = x2 + k, k \{�1}. The proof relies on tools from group theory and probability theory to formulate a sufficient condition for D(P(f, a0)) = 0 in terms of arithmetic properties of the forward orbit of the critical point of f. This provides an analogy to results in real and complex dynamics, where analytic properties of the forward orbit of the critical point have been shown to determine many global dynamical properties of a quadratic polynomial. The article also includes apparently new work on the irreducibility of iterates of quadratic polynomials

Galois theory of iterated endomorphisms

Given an abelian algebraic group A over a global field F, � 2 A(F), and a prime `, the set of all preimages ofunder some iterate of (`) generates an extension of F that contains all `-power torsion points as well as a Kummer-type extension. We analyze the Galois group of this extension, and for several classes of A we give a simple characterization of when the Galois group is as large as possible up to constraints imposed by the endomorphism ring or the Weil pairing. This Galois group encodes information about the density of primes p in the ring of integers of F such that the order of (� mod p) is prime to `. We compute this density in the general case for several classes of A, including elliptic curves and one-dimensional tori. For example, if F is a number field, A=F is an elliptic curve with surjective 2-adic representation and � 2 A(F) with � 㘲 2A(F(A(4))), then the density of p with (� mod p) having odd order is 11=21.

Galois theory of quadratic rational functions

For a number field K with absolute Galois group G_K, we consider the action of G_K on the infinite tree of preimages of a point in K under a degree-two rational function phi, with particular attention to the case when phi commutes with a non-trivial Mobius transfomation. In a sense this is a dynamical systems analogue to the l-adic Galois representation attached to an elliptic curve, with particular attention to the CM case. Using a result about the discriminants of numerators of iterates of phi, we give a criterion for the image of the action to be as large as possible. This criterion is in terms of the arithmetic of the forward orbits of the two critical points of phi. In the case where phi commutes with a non-trivial Mobius transfomation, there is in effect only one critical orbit, and we give a modified version of our maximality criterion. We prove a Serre-type finite-index result in many cases of this latter setting.

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 x 2 + c ∈ Fp[x ], and this in turn is used to establish a basic property of the p-adic Mandelbrot set.

Attracting cycles in p-adic dynamics and height bounds for postcritically finite maps

A rational function of degree at least two with coefficients in an algebraically closed field is post-critically finite (PCF) if all of its critical points have finite forward orbit under iteration. We show that the collection of PCF rational functions is a set of bounded height in the moduli space of rational functions over the complex numbers, once the well-understood family known as flexible Lattes maps is excluded. As a consequence, there are only finitely many conjugacy classes of non-Lattes PCF rational maps of a given degree defined over any given number field. The key ingredient of the proof is a non-archimedean version of Fatous classical result that every attracting cycle of a rational function over the complex numbers attracts a critical point.

Achievement Sets of Sequences

Abstract Given a real sequence (xn), we examine the set of all sums of the form Σi∊Ixi, as I varies over subsets of the positive integers. We call this the achievement set of (xn), and write it AS(xn). For instance, AS(1/2n) = [0, 1] by the existence of binary expansions, and AS(2/3n) is the Cantor middle-third set. We explore the properties of these two sequences that account for their very different achievement sets. We give a sufficient condition for a sequence to have an achievement set that is an interval, and another sufficient condition for the achievement set to be a Cantor set. We also examine what sets can occur as achievement sets, and give results on the topology of achievement sets.

A postmodern view of fractions and the reciprocals of Fermat primes.

In Americas visually-oriented, quantitatively illiterate culture, images have a great deal of power, so if a picture is today worth a thousand words, it must be worth at least a billion numbers. This power of the image is a hallmark of the postmodern era, in which the critical role of the obselver has come to be recognized, and an understanding of the viewpoint has become inseparable from that of the object. In some ways, the blossoming of chaos theoly marked the arrival of mathematical postmodernism. Not so long ago, mathematical ideas were virtually unseen in American popular culture, and it took the enthralling fractal images of chaos theoly to change that: the studies of chaos and fractals became some of the most widely discussed mathematical topics ever, and pictures of fractal images such as the

Eventually stable rational functions

For a field K, rational function phi in K(z) of degree at least two, and alpha in P^1(K), we study the polynomials in K[z] whose roots are given by the solutions to phi^n(z) = alpha, where phi^n denotes the nth iterate of phi. When the number of irreducible factors of these polynomials stabilizes as n grows, the pair (phi, alpha) is called eventually stable over K. We conjecture that (phi, alpha) is eventually stable over K when K is any global field and alpha any point not periodic under phi (an additional non-isotriviality hypothesis is necessary in the function field case). We prove the conjecture when K has a discrete valuation for which (1) phi has good reduction and (2) phi acts bijectively on all finite residue extensions. As a corollary, we prove for these maps a conjecture of Sookdeo on the finiteness of S-integral points in backwards orbits. We also give several characterizations of eventual stability in terms of natural finiteness conditions, and survey previous work on the phenomenon.

Newly reducible iterates in families of quadratic polynomials

For fixed n 3 and nearly all values of , we show that there are only finitely many m such that g; m has a newly reducible n-th iterate. For nD 2 we show a similar result for a much more restricted set of . These results complement those obtained by Danielson and Fein (Proc. Amer. Math. Soc. 130:6 (2002), 1589‐1596) in the higher-degree case. Our method involves translating the problem to one of finding rational points on certain hyperelliptic curves, determining the genus of these curves, and applying Faltings’ theorem.