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John R. Doyle

Publications and source records attributed to John R. Doyle.

26 records · Page 2Linked to original sources
arXiv2017

Dynamical modular curves for quadratic polynomial maps

Motivated by the dynamical uniform boundedness conjecture of Morton and Silverman, specifically in the case of quadratic polynomials, we give a formal construction of a certain class of dynamical analogues of classical modular curves. The preperiodic points for a quadratic polynomial map may be endowed with the structure of a directed graph satisfying certain strict conditions; we call such a graph admissible. Given an admissible graph $G$, we construct a curve $X_1(G)$ whose points parametrize quadratic polynomial maps -- which, up to equivalence, form a one-parameter family -- together with a collection of marked preperiodic points that form a graph isomorphic to $G$. Building on work of Bousch and Morton, we show that these curves are irreducible in characteristic zero, and we give an application of irreducibility in the setting of number fields. We end with a discussion of the Galois theory associated to the preperiodic points of quadratic polynomials, including a certain Galois representation that arises naturally in this setting.

math.DS↗
arXiv2017

Reduction of dynatomic curves

The dynatomic modular curves parametrize polynomial maps together with a point of period $n$. It is known that the dynatomic curves $Y_1(n)$ are smooth and irreducible in characteristic 0 for families of polynomial maps of the form $f_c(z) = z^m +c$ where $m\geq 2$. In the present paper, we build on the work of Morton to partially characterize the primes $p$ for which the reduction modulo $p$ of $Y_1(n)$ remains smooth and/or irreducible. As an application, we give new examples of good reduction of $Y_1(n)$ for several primes dividing the ramification discriminant when $n=7,8,11$. The proofs involve arithmetic and complex dynamics, reduction theory for curves, ramification theory, and the combinatorics of the Mandelbrot set.

math.DS↗
arXiv2016

Preperiodic portraits for unicritical polynomials

Let $K$ be an algebraically closed field of characteristic zero, and for $c \in K$ and an integer $d \ge 2$, define $f_{d,c}(z) := z^d + c \in K[z]$. We consider the following question: If we fix $x \in K$ and integers $M \ge 0$, $N \ge 1$, and $d \ge 2$, does there exist $c \in K$ such that, under iteration by $f_{d,c}$, the point $x$ enters into an $N$-cycle after precisely $M$ steps? We conclude that the answer is generally affirmative, and we explicitly give all counterexamples. When $d = 2$, this answers a question posed by Ghioca, Nguyen, and Tucker.

math.DS↗
arXiv2016

Preperiodic portraits for unicritical polynomials over a rational function field

Let $K$ be an algebraically closed field of characteristic zero, and let $\mathcal{K} := K(t)$ be the rational function field over $K$. For each $d \ge 2$, we consider the unicritical polynomial $f_d(z) := z^d + t \in \mathcal{K}[z]$, and we ask the following question: If we fix $α\in \mathcal{K}$ and integers $M \ge 0$, $N \ge 1$, and $d \ge 2$, does there exist a place $\mathfrak{p} \in \operatorname{Spec} K[t]$ such that, modulo $\mathfrak{p}$, the point $α$ enters into an $N$-cycle after precisely $M$ steps under iteration by $f_d$? We answer this question completely, concluding that the answer is generally affirmative and explicitly giving all counterexamples. This extends previous work by the author in the case that $α$ is a constant point.

math.DS↗
arXiv2015

Configuration of the Crucial Set for a Quadratic Rational Map

Let $K$ be a complete, algebraically closed non-archimedean valued field, and let $φ(z) \in K(z)$ have degree two. We describe the crucial set of $φ$ in terms of the multipliers of $φ$ at the classical fixed points, and use this to show that the crucial set determines a stratification of the moduli space $\mathcal{M}_2(K)$ related to the reduction type of $φ$. We apply this to settle a special case of a conjecture of Hsia regarding the density of repelling periodic points in the non-archimedean Julia set.

math.NT↗
arXiv2014

Preperiodic points for quadratic polynomials over quadratic fields

To each quadratic number field $K$ and each quadratic polynomial $f$ with $K$-coefficients, one can associate a finite directed graph $G(f,K)$ whose vertices are the $K$-rational preperiodic points for $f$, and whose edges reflect the action of $f$ on these points. This paper has two main goals. (1) For an abstract directed graph $G$, classify the pairs $(K,f)$ such that the isomorphism class of $G$ is realized by $G(f,K)$. We succeed completely for many graphs $G$ by applying a variety of dynamical and Diophantine techniques. (2) Give a complete description of the set of isomorphism classes of graphs that can be realized by some $G(f,K)$. A conjecture of Morton and Silverman implies that this set is finite. Based on our theoretical considerations and a wealth of empirical evidence derived from an algorithm that is developed in this paper, we speculate on a complete list of isomorphism classes of graphs that arise from quadratic polynomials over quadratic fields.

math.NT↗
arXiv2012

Apollonian circle packings of the half-plane

We consider Apollonian circle packings of a half Euclidean plane. We give necessary and sufficient conditions for two such packings to be related by a Euclidean similarity (that is, by translations, reflections, rotations and dilations) and describe explicitly the group of self-similarities of a given packing. We observe that packings with a non-trivial self-similarity correspond to positive real numbers that are the roots of quadratic polynomials with rational coefficients. This is reflected in a close connection between Apollonian circle packings and continued fractions which allows us to completely classify such packings up to similarity.

math.MG↗