arXiv subjects

Ophelia Adams

Publications and source records attributed to Ophelia Adams.

5 recordsLinked to original sources

arXiv2025

Semiconjugacy and Self-Similar Subgroups of pfIMGs

The profinite iterated monodromy group (pfIMG) is a self-similar group associated to dynamical systems. We show that its proper open self-similar subgroups correspond to highly rigid semiconjugacies, which we partly classify in general. For polynomials, we show that only the twisted Chebyshev maps can arise. Next, we define and construct self-similar closures of subgroups of pfIMGs, and show that this preserves many group-theoretic properties of the original subgroup. As a consequence, we conclude that pfIMGs with open subgroups satisfying certain properties (e.g. prosolvable or pronilpotent) either satisfy that property themselves, or arise from one of these exceptional semiconjugacies. This is applied to answer some questions posed in [BGJT25] about open Frattini subgroups of pfIMGs: unicritical polynomials of composite degree do not have an open Frattini subgroup, and a polynomial with an open Frattini subgroup is often pro-$p$.

math.NT↗

arXiv2025

Profinite Iterated Monodromy Groups of Unicritical Polynomials

Let $f(x) = ax^d + b \in K[x]$ be a unicritical polynomial with degree $d \geq 2$ which is coprime to $\mathrm{char} K$. We provide an explicit presentation for the profinite iterated monodromy group of $f$, analyze the structure of this group, and use this analysis to determine the constant field extension in $K(f^{-\infty}(t))/K(t)$.

math.NT↗

arXiv2022

A Dynamical Analogue of the Criterion of N\'eron-Ogg-Shafarevich

We introduce an anabelian approach to the study of arboreal Galois representations and apply Tamagawa's anabelian version of the N\'eron-Ogg-Shafarevich criterion to produce a dynamical analogue of this criterion: unramified representations correspond to rational maps satisfying a strong form of good reduction in terms of their critical locus. Subsequently, we pursue a dynamical anlaogue of the N\'eron-Ogg-Shafarevich criterion in terms of the more (dynamically) traditional arboreal representations, which relates unramified arboreal representations to a certain separability condition on the dynamical system. Finally, we relate the our criteria: the anabelian criterion corresponds to the dynamical criterion as one varies the base point around the critical locus. Along the way we develop effective criteria to determine which primes are infinitely ramified in arboreal representations over number fields, as well as the asymptotic growth of that ramification; we conclude with examples and applications, especially to dynamical systems over number fields.

math.NT↗

arXiv2021

A Dynamical Analogue of Sen's Theorem

We study the higher ramification structure of dynamical branch extensions, and propose a connection between the natural dynamical filtration and the filtration arising from the higher ramification groups: each member of the former should, after a linear change of index, coincide with a member of the latter. This is an analogue of Sen's theorem on ramification in $p$-adic Lie extensions. By explicitly calculating the Hasse-Herbrand functions of such branch extensions, we are able to show that this description is accurate for some families of polynomials, in particular post-critically bounded polynomials of $p$-power degree. We apply our results to give a partial answer to a question of Berger (in arXiv:1411.7064) and a partial answer to a question about wild ramification in arboreal extensions of number fields (raised in both arXiv:math/0408170 and arXiv:1511.00194).

math.NT↗

arXiv2017

Conditions for Solvability in Chemical Reaction Networks at Quasi-Steady-State

The quasi-steady-state assumption (QSSA) is an approximation that is widely used in chemistry and chemical engineering to simplify reaction mechanisms. The key step in the method requires a solution by radicals of a system of multivariate polynomials. However, Pantea, Gupta, Rawlings, and Craciun showed that there exist mechanisms for which the associated polynomials are not solvable by radicals, and hence reduction by QSSA is not possible. In practice, however, reduction by QSSA always succeeds. To provide some explanation for this phenomenon, we prove that solvability is guaranteed for a class of common chemical reaction networks. In the course of establishing this result, we examine the question of when it is possible to ensure that there are finitely many (quasi) steady states. We also apply our results to several examples, in particular demonstrating the minimality of the nonsolvable example presented by Pantea, Gupta, Rawlings, and Craciun.

math.DS↗