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Rafe Jones

Publications and source records attributed to Rafe Jones.

At least 19 recordsLinked to original sources
arXiv2026

2007: An Arboreal Odyssey: A View of Arboreal Galois Representations and Applications, from Early in the Subject's History

The study of arboreal Galois representations (that is, Galois groups arising from iteration of polynomial and rational functions) originated with work of Odoni in the 1980s. Beginning in the early 2000s it underwent a period of renewed interest, which continues to this day. Written in 2007, this survey article gives a sense of the subject from the early days of this renewal. It is presented here as a document of historical interest -- precisely as originally written -- and because some recent work has referenced specific pieces of it. It was written as an informal document, and not intended to be published. Much, though not all, of the content overlaps with the 2013 survey article ``Galois representations from pre-image trees: an arboreal survey" of the author.

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arXiv2026

On the Arithmetic of Bicritical Rational Functions

Bicritical rational functions -- those with precisely two critical points -- include the well-studied families of unicritical polynomials and quadratic rational functions. In this article we lay out general foundations for studying arithmetic dynamical properties of bicritical rational functions, and prove new Galois-theoretic results for a family with special properties. We study the field of definition of the critical points, and give a normal form up to M\"obius conjugacy over this field. As a corollary, we show that after a finite extension of the ground field, the arboreal Galois representation attached to a bicritical rational function injects into an iterated wreath product of cyclic groups. We then examine the family of quadratic $\phi \in \mathbb{Q}(x)$ with critical points $\gamma_1$ and $\gamma_2$ such that $\phi(\gamma_1) = \gamma_2$. Adapting methods of Odoni-Stoll in the polynomial case to rational functions, we show that the arboreal representation is surjective for an infinite subfamily.

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arXiv2022

Roots of unity and higher ramification in iterated extensions

Given a field $K$, a rational function $\phi \in K(x)$, and a point $b \in \mathbb{P}^1(K)$, we study the extension $K(\phi^{-\infty}(b))$ generated by the union over $n$ of all solutions to $\phi^n(x) = b$, where $\phi^n$ is the $n$th iterate of $\phi$. We ask when a finite extension of $K(\phi^{-\infty}(b))$ can contain all $m$-power roots of unity for some $m \geq 2$, and prove that several families of rational functions do so. A motivating application is to understand the higher ramification filtration when $K$ is a finite extension of $\mathbb{Q}_p$ and $p$ divides the degree of $\phi$, especially when $\phi$ is post-critically finite (PCF). We show that all higher ramification groups are infinite for new families of iterated extensions, for example those given by bicritical rational functions with periodic critical points. We also give new examples of iterated extensions with subextensions satisfying an even stronger ramification-theoretic condition called arithmetic profiniteness. We conjecture that every iterated extension arising from a PCF map should have a subextension with this stronger property, which would give a dynamical analogue of Sen's theorem for PCF maps.

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arXiv2021

Galois groups and prime divisors in random quadratic sequences

Given a set $S=\{x^2+c_1,\dots,x^2+c_s\}$ defined over a field and an infinite sequence $\gamma$ of elements of $S$, one can associate an arboreal representation to $\gamma$, generalizing the case of iterating a single polynomial. We study the probability that a random sequence $\gamma$ produces a ``large-image'' representation, meaning that infinitely many subquotients in the natural filtration are maximal. We prove that this probability is positive for most sets $S$ defined over $\mathbb{Z}[t]$, and we conjecture a similar positive-probability result for suitable sets over $\mathbb{Q}$. As an application of large-image representations, we prove a density-zero result for the set of prime divisors of some associated quadratic sequences. We also consider the stronger condition of the representation being finite-index, and we classify all $S$ possessing a particular kind of obstruction that generalizes the post-critically finite case in single-polynomial iteration.

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arXiv2021

Iterated monodromy groups of rational functions and periodic points over finite fields

Let $q$ be a prime power and $\phi$ a rational function with coefficients in a finite field $\mathbb{F}_q$. For $n \geq 1$, each element of $\mathbb{P}^1(\F_{q^n})$ is either periodic or strictly preperiodic under iteration of $\phi$. Denote by $a_n$ the proportion of periodic elements. Little is known about how $a_n$ changes as $n$ grows, unless $\phi$ is a power map or Chebyshev polynomial. We give the first results on this question for a wider class of rational functions: $a_n$ has lim inf $0$ when $q$ is odd and $\phi$ is quadratic and neither Latt\`es nor conjugate to a one-parameter family of exceptional maps. We also show that $a_n$ has limit $0$ when $\phi$ is a non-Chebyshev quadratic polynomial with strictly preperiodic finite critical point and $q$ is an odd square. Our methods yield additional results on periodic points for reductions of post-critically finite (PCF) rational functions defined over number fields. The difficulty of understanding $a_n$ in general is that $\mathbb{P}^1(\F_{q^n})$ is a finite set with no ambient geometry. In fact, $\phi$ can be lifted to a PCF rational map on the Riemann sphere, where we show that $a_n$ is given by counting elements of the iterated monodromy group (IMG) that act with fixed points at all levels of the tree of preimages. Using a martingale convergence theorem, we translate the problem to determining whether certain IMG elements exist. This in turn can be decisively addressed using the expansion of PCF rational maps in the orbifold metric.

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arXiv2020

Newly reducible polynomial iterates

Given a field $K$ and $n > 1$, we say that a polynomial $f \in K[x]$ has newly reducible $n$th iterate over $K$ if $f^{n-1}$ is irreducible over $K$, but $f^n$ is not (here $f^i$ denotes the $i$th iterate of $f$). We pose the problem of characterizing, for given $d,n > 1$, fields $K$ such that there exists $f \in K[x]$ of degree $d$ with newly reducible $n$th iterate, and the similar problem for fields admitting infinitely many such $f$. We give results in the cases $(d,n) \in \{(2,2), (2,3), (3,2), (4,2)\}$ as well as for $(d,2)$ when $d \equiv 2 \bmod{4}$. In particular, we show that for all these $(d,n)$ pairs, there are infinitely many monic $f \in \mathbb{Z}[x]$ of degree $d$ with newly reducible $n$th iterate over $\mathbb{Q}$. Curiously, the minimal polynomial $x^2 - x - 1$ of the golden ratio is one example of $f \in \mathbb{Z}[x]$ with newly reducible third iterate; very few other examples have small coefficients. Our investigations prompt a number of conjectures and open questions.

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arXiv2019

Eventually stable quadratic polynomials over $\mathbb{Q}$

We study the number of irreducible factors (over $\mathbb{Q}$) of the $n$th iterate of a polynomial of the form $f_r(x) = x^2 + r$ for rational $r$. When the number of such factors is bounded independent of $n$, we call $f_r(x)$ \textit{eventually stable} (over $\mathbb{Q}$). Previous work of Hamblen, Jones, and Madhu shows that $f_r$ is eventually stable unless $r$ has the form $1/c$ for some integer $c \not\in \{0,-1\}$, in which case existing methods break down. We study this family, and prove that several conditions on $c$ of various flavors imply that all iterates of $f_{1/c}$ are irreducible. We give an algorithm that checks the latter property for all $c$ up to a large bound $B$ in time polynomial in $\log B$. We find all $c$-values for which the third iterate of $f_{1/c}$ has at least four irreducible factors, and all $c$-values such that $f_{1/c}$ is irreducible but its third iterate has at least three irreducible factors. This last result requires finding all rational points on a genus-2 hyperelliptic curve for which the method of Chabauty and Coleman does not apply; we use the more recent variant known as elliptic Chabauty. Finally, we apply all these results to completely determine the number of irreducible factors of any iterate of $f_{1/c}$, for all $c$ with absolute value at most $10^9$.

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arXiv2018

Current Trends and Open Problems in Arithmetic Dynamics

Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry, and partly from $p$-adic analogues of theorems and conjectures in classical complex dynamics. In this article we survey some of the motivating problems and some of the recent progress in the field of arithmetic dynamics.

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arXiv2017

Riccati equations and polynomial dynamics over function fields

Given a function field $K$ and $\phi \in K[x]$, we study two finiteness questions related to iteration of $\phi$: whether all but finitely many terms of an orbit of $\phi$ must possess a primitive prime divisor, and whether the Galois groups of iterates of $\phi$ must have finite index in their natural overgroup $\text{Aut}(T_d)$, where $T_d$ is the infinite tree of iterated preimages of $0$ under $\phi$. We focus particularly on the case where $K$ has characteristic $p$, where far less is known. We resolve the first question in the affirmative under relatively weak hypotheses; interestingly, the main step in our proof is to rule out "Riccati differential equations" in backwards orbits. We then apply our result on primitive prime divisors and adapt a method of Looper to produce a family of polynomials for which the second question has an affirmative answer; these are the first non-isotrivial examples of such polynomials. We also prove that almost all quadratic polynomials over $\mathbb{Q}(t)$ have iterates whose Galois group is all of $\text{Aut}(T_d)$.

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arXiv2016

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.

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arXiv2015

Powers in orbits of rational functions: cases of an arithmetic dynamical Mordell-Lang conjecture

Let $K$ be a finitely generated field of characteristic zero. We study, for fixed $m \geq 2$, the rational functions $\phi$ defined over $K$ that have a $K$-orbit containing infinitely many distinct $m$th powers. For $m \geq 5$ we show the only such functions are those of the form $cx^j(\psi(x))^m$ with $\psi \in K(x)$, and for $m \leq 4$ we show the only additional cases are certain Latt\`es maps and four families of rational functions whose special properties appear not to have been studied before. With additional analysis, we show that the index set $\{n \geq 0 : \phi^{n}(a) \in \lambda(\mathbb{P}^1(K))\}$ is a union of finitely many arithmetic progressions, where $\phi^{n}$ denotes the $n$th iterate of $\phi$ and $\lambda \in K(x)$ is any map M\"obius-conjugate over $K$ to $x^m$. When the index set is infinite, we give bounds on the number and moduli of the arithmetic progressions involved. These results are similar in flavor to the dynamical Mordell-Lang conjecture, and motivate a new conjecture on the intersection of an orbit with the value set of a morphism. A key ingredient in our proofs is a study of the curves $y^m = \phi^{n}(x)$. We describe all $\phi$ for which these curves have an irreducible component of genus at most 1, and show that such $\phi$ must have two distinct iterates that are equal in $K(x)^*/K(x)^{*m}$.

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arXiv2015

Finite ramification for preimage fields of postcritically finite morphisms

Given a finite endomorphism $\varphi$ of a variety $X$ defined over the field of fractions $K$ of a Dedekind domain, we study the extension $K(\varphi^{-\infty}(\alpha)) : = \bigcup_{n \geq 1} K(\varphi^{-n}(\alpha))$ generated by the preimages of $\alpha$ under all iterates of $\varphi$. In particular when $\varphi$ is post-critically finite, i.e., there exists a non-empty, Zariski-open $W \subseteq X$ such that $\varphi^{-1}(W) \subseteq W$ and $\varphi : W \to X$ is \'etale, we prove that $K(\varphi^{-\infty}(\alpha))$ is ramified over only finitely many primes of $K$. This provides a large supply of infinite extensions with restricted ramification, and generalizes results of Aitken-Hajir-Maire in the case $X = \mathbb{A}^1$ and Cullinan-Hajir, Jones-Manes in the case $X = \mathbb{P}^1$. Moreover, we conjecture that this finite ramification condition characterizes post-critically finite morphisms, and we give an entirely new result showing this for $X = \mathbb{P}^1$. The proof relies on Faltings' theorem and a local argument.

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arXiv2015

Wild ramification in a family of low-degree extensions arising from iteration

This article gives a first look at wild ramification in a family of iterated extensions. For integer values of c, we consider the splitting field of $(x^2 + c)^2 + c$, the second iterate of $x^2 + c$. We give complete information on the factorization of the ideal (2) as c varies, and find a surprisingly complicated dependence of this factorization on the parameter c. We show that 2 ramifies (necessarily wildly) in all these extensions except when c = 0, and we describe the higher ramification groups in some totally ramified cases.

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arXiv2014

Galois representations from pre-image trees: an arboreal survey

Given a global field K and a rational function phi defined over K, one may take pre-images of 0 under successive iterates of phi, and thus obtain an infinite rooted tree T by assigning edges according to the action of phi. The absolute Galois group of K acts on T by tree automorphisms, giving a subgroup G(phi) of the group Aut(T) of all tree automorphisms. Beginning in the 1980s with work of Odoni, and developing especially over the past decade, a significant body of work has emerged on the size and structure of this Galois representation. These inquiries arose in part because knowledge of G(phi) allows one to prove density results on the set of primes of K that divide at least one element of a given orbit of phi. Following an overview of the history of the subject and two of its fundamental questions, we survey cases where G(phi) is known to have finite index in Aut(T). While it is tempting to conjecture that such behavior should hold in general, we exhibit four classes of rational functions where it does not, illustrating the difficulties in formulating the proper conjecture. Fortunately, one can achieve the aforementioned density results with comparatively little information about G(phi), thanks in part to a surprising application of probability theory. Underlying all of this analysis are results on the factorization into irreducibles of the numerators of iterates of phi, which we survey briefly. We find that for each of these matters, the arithmetic of the forward orbits of the critical points of phi proves decisive, just as the topology of these orbits is decisive in complex dynamics.

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arXiv2013

Attracting cycles in p-adic dynamics and height bounds for post-critically 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 Fatou's classical result that every attracting cycle of a rational function over the complex numbers attracts a critical point.

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arXiv2013

The density of primes in orbits of z^d + c

Given a polynomial f(z) = z^d + c over a global field K and a_0 in K, we study the density of prime ideals of K dividing at least one element of the orbit of a_0 under f. The density of such sets for linear polynomials has attracted much study, and the second author has examined several families of quadratic polynomials, but little is known in the higher-degree case. We show that for many choices of d and c this density is zero for all a_0, assuming K contains a primitive dth root of unity. The proof relies on several new results, including some ensuring the number of irreducible factors of the nth iterate of f remains bounded as n grows, and others on the ramification above certain primes in iterated extensions. Together these allow for nearly complete information when K is a global function field or when K=Q(zeta_d).

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arXiv2012

Newly reducible iterates in families of quadratic polynomials

We examine the question of when a quadratic polynomial f(x) defined over a number field K can have a newly reducible nth iterate, that is, f^n(x) irreducible over K but f^{n+1}(x) reducible over K, where f^n denotes the nth iterate of f. For each choice of critical point \gamma of f(x), we consider the family g_{\gamma,m}(x)= (x - \gamma)^2 + m + \gamma, m \in K. For fixed n \geq 3 and nearly all values of \gamma, we show that there are only finitely many m such that g_{\gamma,m} has a newly reducible nth iterate. For n = 2 we show a similar result for a much more restricted set of \gamma. These results complement those obtained by Danielson and Fein 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.

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