106 citations · 200 across the 16 of their papers we have counts for
25 papers
2007: An Arboreal Odyssey: A View of Arboreal Galois Representations and Applications, from Early in the Subject's History
Rafe Jones
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. Beginn…
Corrigendum and addendum to "Fixed-point-free elements of iterated monodromy groups" [Trans. of the American Mathematical Society Vol. 367, No. 3 (2015), 2023-2049]
Jorge Fariña-Asategui, Rafe Jones, Santiago Radi
In this note we document a gap in an argument in the above paper, and point to new work in the literature giving a complete proof of the main result.
On the Arithmetic of Bicritical Rational Functions
Vefa Goksel, Rafe Jones
Bicritical rational functions -- those with precisely two critical points -- include the well-studied families of unicritical polynomials and quadratic rational functions. In this…
Roots of unity and higher ramification in iterated extensions
Spencer Hamblen, Rafe Jones
Given a field , a rational function , and a point , we study the extension generated by the union over of all solutions…
Galois groups and prime divisors in random quadratic sequences
John R. Doyle, Vivian Olsiewski Healey, Wade Hindes +1
Given a set defined over a field and an infinite sequence of elements of , one can associate an arboreal representation to , generalizing th…
Iterated monodromy groups of rational functions and periodic points over finite fields
Andrew Bridy, Rafe Jones, Gregory Kelsey +1
Let be a prime power and a rational function with coefficients in a finite field . For , each element of $\mathbb{P}^1(\F_{q^n})$ is either periodic…
Newly reducible polynomial iterates
Peter Illig, Rafe Jones, Eli Orvis +2
Given a field and , we say that a polynomial has newly reducible th iterate over if is irreducible over , but is not (here d…
Eventually stable quadratic polynomials over
David DeMark, Wade Hindes, Rafe Jones +3
We study the number of irreducible factors (over ) of the th iterate of a polynomial of the form for rational . When the number of such factors…
Current Trends and Open Problems in Arithmetic Dynamics
Robert Benedetto, Laura DeMarco, Patrick Ingram +4
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 c…
Riccati equations and polynomial dynamics over function fields
Wade Hindes, Rafe Jones
Given a function field and , we study two finiteness questions related to iteration of : whether all but finitely many terms of an orbit of must possess a pri…
Eventually stable rational functions
Rafe Jones, Alon Levy
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) = alph…
Finite ramification for preimage fields of postcritically finite morphisms
Andrew Bridy, Patrick Ingram, Rafe Jones +5
Given a finite endomorphism of a variety defined over the field of fractions of a Dedekind domain, we study the extension $K(φ^{-\infty}(α)) : = \bigcup_{n \geq 1} K(φ^…
Powers in orbits of rational functions: cases of an arithmetic dynamical Mordell-Lang conjecture
Jordan Cahn, Rafe Jones, Jacob Spear
Let be a finitely generated field of characteristic zero. We study, for fixed , the rational functions defined over that have a -orbit containing infinitel…
Wild ramification in a family of low-degree extensions arising from iteration
Benjamin Breen, Rafe Jones, Tommy Occhipinti +1
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 , the second i…
Galois representations from pre-image trees: an arboreal survey
Rafe Jones
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 assig…
The density of primes in orbits of z^d + c
Spencer Hamblen, Rafe Jones, Kalyani Madhu
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 densi…
Newly reducible iterates in families of quadratic polynomials
Katharine Chamberlin, Emma Colbert, Sharon Frechette +3
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)…
Fixed-point-free elements of iterated monodromy groups
Rafe Jones
The iterated monodromy group of a post-critically finite complex polynomial of degree d \geq 2 acts naturally on the complete d-ary rooted tree T of preimages of a generic point. T…
Attracting cycles in p-adic dynamics and height bounds for post-critically finite maps
Robert L. Benedetto, Patrick Ingram, Rafe Jones +1
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 orbi…
Galois theory of quadratic rational functions
Rafe Jones, Michelle Manes
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…
An iterative construction of irreducible polynomials reducible modulo every prime
Rafe Jones
We give a method of constructing polynomials of arbitrarily large degree irreducible over a global field F but reducible modulo every prime of F. The method consists of finding qua…
Blocks of monodromy groups in Complex Dynamics
Rafe Jones, Han Peters
Motivated by a problem in complex dynamics, we examine the block structure of the natural action of monodromy groups on the tree of preimages of a generic point. We show that in ma…
Uniform Bounds on Pre-Images under Quadratic Dynamical Systems
Xander Faber, Benjamin Hutz, Patrick Ingram +4
For any elements b,c of a number field K, let G(b,c) denote the backwards orbit of b under the map f_c: C-->C given by f_c(x)=x^2+c. We prove an upper bound on the number of elemen…
Galois theory of iterated endomorphisms
Rafe Jones, Jeremy Rouse
Given an abelian algebraic group over a global field , , and a prime , the set of all preimages of under some iterate of generates an extension…