AMS Special Session on Arithmetic Dynamics (SS 5A)

Joint Mathematics Meetings (JMM)
Jan. 15–18, 2020 (Weds–Sat), Denver, Colorado
Denver Convention Center, Room 303

Organizers: Rafe Jones, Nicole Looper, Joe Silverman

Talk # Date Time Speaker Title Slides (if available)
Friday morning, January 17, 2020
1 Fri AM 8:00am Keping Huang Uniform Bounds for Periods of Endomorphisms of Varieties 1154-11-137
2 Fri AM 8:30am Xander Faber Totally $T$-adic functions of small height 1154-11-393
3 Fri AM 9:00am Bella Tobin Post-Critically Finite Bicritical Polynomials over Number Fields 1154-11-1619
4 Fri AM 9:30am Sarah Koch Multiplier spectra of PCF rational maps 1154-37-1328
5 Fri AM 10:00am Trevor Hyde Portrait spaces for dynamical semigroups and unlikely intersections 1154-37-1015
6 Fri AM 10:30am Patrick Ingram A dynamical Néron symbol 1154-11-653
Friday afternoon, January 17, 2020
7 Fri PM 1:00pm Robin Zhang The Galois-dynamics correspondence for unicritical polynomials 1154-11-1831
8 Fri PM 1:30pm Andrew Bridy The Arakelov-Zhang pairing and Julia sets 1154-11-1840
9 Fri PM 2:00pm Olga Lukina Arboreal Cantor actions 1154-37-215
10 Fri PM 2:30pm Vefa Goksel Misiurewicz polynomials and irreducibility 1154-11-172
11 Fri PM 3:00pm Jason Bell A gap conjecture for heights of iterates 1154-11-479
12 Fri PM 3:30pm Jeffrey Diller A transcendental dynamical degree 1154-32-999
13 Fri PM 4:00pm Thomas Silverman A non-archimedean $\lambda$-lemma and $J$-stability 1154-37-1539
14 Fri PM 4:30pm Wade Hindes Dynamical height growth: left, right, and total orbits 1154-11-130
15 Fri PM 5:00pm Rob Benedetto Isolation of postcritically finite parameters in $p$-adic dynamical moduli spaces 1154-11-221
16 Fri PM 5:30pm Michelle Manes Post-critically finite cubic polynomials 1154-11-2083
Saturday morning, January 18, 2020
17 Sat AM 8:00am Andrea Ferraguti The inverse problem for arboreal Galois representations of index two 1154-11-395
18 Sat AM 8:30am John Lesieutre Higher arithmetic degrees of rational maps 1154-14-831
19 Sat AM 9:00am Matthew Satriano New types of heights with connections to the Batyrev-Manin and Malle Conjectures 1154-11-123
20 Sat AM 9:30am Yohsuke Matsuzawa On Kawaguchi-Silverman conjecture for endomorphisms of rationally connected varieties 1154-14-1622
21 Sat AM 10:00am Ben Hutz Automorphism loci for endomorphisms of $\mathbb{P}^1$ 1154-37-1621
22 Sat AM 10:30am Jamie Juul Arboreal Galois representations of PCF quadratic polynomials 1154-11-1924
23 Sat AM 11:00am David Krumm Algebraic preperiodic points of entire transcendental functions 1154-37-818
24 Sat AM 11:30am John Doyle Moduli spaces for dynamical systems with level structure 1154-37-668

Titles and Abstracts

FRIDAY MORNING

1. Fri 8:00am. 1154-11-137
Keping Huang
Uniform Bounds for Periods of Endomorphisms of Varieties. Preliminary report.
Abstract. Suppose $X$ is a projective variety defined over a finite extension $K$ of $\mathbb Q_p$ and suppose $X$ admits a model $\mathcal X$ defined over the ring of integers $R$ of $K$. Let $f:X\to X$ be an endomorphism of $X$ defined over $K$ that can be extended to an endomorphism of $\mathcal X$ defined over $R$. We prove an upper bound for the primitive period of periodic points defined over $R$.
2. Fri 8:30am. 1154-11-393
Xander Faber
Totally $T$-adic functions of small height.
Abstract. A nonzero algebraic number $\alpha$ is totally $p$-adic if its minimal polynomial (over $\mathbb Q$) splits completely over $\mathbb Q_p$. If $\alpha$ is not a $(p-1)$st root of unity, then the naive logarithmic height of such an element is uniformly bounded away from zero by an equidistribution result of Bombieri/Zannier or an elementary inequality of Pottmeyer. In this work, we introduce a geometric analogue. Fix a finite field $\mathbb F_q$, and consider the rational function field $\mathbb F_q(T)$. An algebraic function $f$ that generates a separable extension of $\mathbb F_q(T)$ is totally $T$-adic if its minimal polynomial (over $\mathbb F_q(T)$) splits completely in the field of Laurent series $\mathbb F_q((T))$. We will discuss a lower bound for the height of any nonconstant totally $T$-adic function, and we will show that functions achieving the lower bound give rise to curious algebraic curves over $\mathbb F_q$ with many rational points. We also investigate the limit-infimum of the heights of totally $T$-adic functions using a dynamical construction.
3. Fri 9:00am. 154-11-1619
Bella Tobin
Post-Critically Finite Bicritical Polynomials over Number Fields. Preliminary report.
Abstract. Preliminary report. Unicritical polynomials have been studied extensively in dynamics. A natural next step is to consider polynomials with two finite critical points, namely bicritical polynomials. I present results on the family of bicritical polynomials over a number field $K$, including results on post-critically finite bicritical polynomials over $\mathbb Q$.
4. Fri 9:30am. 1154-37-1328
Sarah Koch
Multiplier spectra of PCF rational maps.
Abstract. We discuss the question of which algebraic numbers arise as multipliers of PCF rational maps, and to what extent this data determines the rational map in moduli space. This is very much work in progress, with X. Buff and A. Epstein.
5. Fri 10:00am. 1154-37-1015
Trevor Hyde
Portrait spaces for dynamical semigroups and unlikely intersections. Preliminary report.
Abstract. Given a dynamical portrait for several rational functions acting on a finite set, we initiate the study of the corresponding moduli space of realizations. In this talk we discuss the various ways in which these moduli spaces can have higher than expected dimension and present results explaining this phenomenon. Time permitting we will also survey the findings of large exhaustive computations we conducted for portrait spaces of cubic polynomials.
6. Fri 10:30am. 1154-11-653
Patrick Ingram
A dynamical Néron symbol. Preliminary report.
Abstract. We will discuss a local Néron symbol for arithmetic dynamics, which pairs points and divisors on projective space relative to a given endomorphism, with applications to computing the canonical and critical heights.

FRIDAY AFTERNOON

7. Fri 1:00pm. 1154-11-1831
Robin Zhang
The Galois-dynamics correspondence for unicritical polynomials. Preliminary report.
Abstract. We study a correspondence between Galois actions and dynamical actions on periodic points of the polynomial $\phi(z) =z^d+c$ with $d$ an integer greater than 1 and $c$ a rational number. In particular, this correspondence exists for almost all rational $c$ by a form of Hilbert's irreducibility theorem. When $K$ is a quadratic number field and $d= 2$, this correspondence gives a criterion for the nonexistence of $K$-rational 5-cycles of $z^2+c$ and for the complete determination of $K$-rational 6-cycles of $z^2+c$.
8. Fri 1:30pm. 1154-11-1840
Andrew Bridy
The Arakelov-Zhang pairing and Julia sets.
Abstract. The Arakelov-Zhang pairing $\langle\psi,\phi\rangle$ is a measure of the dynamical distance between two rational maps $\psi$ and $\phi$ over a number field $K$, defined in terms of local integrals on Berkovich space at each completion of $K$. We obtain a simple expression for the important case of the pairing with a power map, which may be interpreted as a limiting height of generic preimages. The expression is in terms of integrals over Julia sets; under certain disjointness conditions on Julia sets, it simplifies to a single canonical height term (in general, this term is a lower bound). This is joint work with Matt Larson.
9. Fri 2:00pm. 1154-37-215
Olga Lukina
Arboreal Cantor actions. Preliminary report.
Abstract. Given an arboreal representation of the absolute Galois group of a field, we associate to it an action of a countably generated discrete group on a Cantor set. We then classify certain classes of arboreal representations by their topological dynamical properties.
10. Fri 2:300pm. 1154-11-172
Vefa Goksel
Misiurewicz polynomials and irreducibility. Preliminary report.
Abstract. Let $f_{c,d}(x)=x^d+c\in\mathbb C[x]$. The $c_0$ values for which $f_{c_0,d}$ has a strictly pre-periodic finite critical orbit are called Misiurewicz points. Any Misiurewicz point lies in $\overline{\mathbb Q}$ Suppose that the Misiurewicz points $c_0,c_1\in\overline{\mathbb Q}$ are such that the polynomials $f_{c_0,d}$ and $f_{c_1,d}$ have the same orbit type. One classical question is whether $c_0$ and $c_1$ need to be Galois conjugates or not. I will talk about some partial results I have recently obtained related to this question.
11. Fri 3:00pm. 1154-11-479
Jason P Bell
A gap conjecture for heights of iterates.
Abstract. Let $X$ be a quasi-projective variety defined over a field $K$ of characteristic 0, endowed with the action of an étale endomorphism $\Phi$, and $f:X\to Y$ is a morphism with $Y$ a quasi-projective variety defined over $K$. Then we show that a uniform result of the following type holds: if for a given $x\in X(K)$, for each $y\in Y(K)$ the set $$S_y:=\{n\in \mathbb N:f(\Phi^n(x)) =y\}$$ is finite, then there exists a positive integer $N$ such that $\#S_y\le N$ for each $y\in Y(K)$. We use this to prove that a "gap" theorem holds for étale endomorphisms, which we now describe. Let $K$ be a number field, $f:X\to\mathbb P^1$ a rational map, and $\Phi$ be an étale endomorphism of $X$. If $\mathcal O$ denotes the forward orbit of $x$ under the action of $\Phi$, then either $f(\mathcal O)$ is finite, or $$ \limsup_{n\to\infty} \frac{h(f(\Phi^n(x)))}{\log n} > 0, $$ where $h(\cdot)$ represents the usual logarithmic Weil height for algebraic points. We conjecture that this dichotomy should hold for general endomorphisms of quasi-projective varieties when everything is defined over a number field.
12. Fri 3:30pm. 1154-32-999
Jeffrey Diller
A transcendental dynamical degree.
Abstract. I will discuss an example of a rational self-map of the projective plane whose first dynamical degree turns out to be transcendental.
13. Fri 4:00pm. 1154-37-1539
Thomas Silverman
A non-archimedean $\lambda$-lemma and $J$-stability.
Abstract. In a celebrated paper published in 1983, R. Mãné, P. Sad, and D. Sullivan prove a result about holomorphic families of injections called the $\lambda$-Lemma with impressive applications to the complex dynamics of families of one-variable rational functions. In this talk, I will discuss the dynamics of families of one-variable rational functions parametrized by Berkovich spaces over a complete non-archimedean field, including a suitable non-archimedean analogue of the &\lambda$-Lemma. I will also explain how this can be used to prove the equivalence of two stability conditions in non-archimedean dynamics.
14. Fri 4:30pm. 1154-11-130
Wade Hindes
Dynamical height growth: left, right, and total orbits. Preliminary report.
Abstract. Let $S$ be a set of dominant rational self-maps on $\mathbb P^N$. We study the arithmetic and dynamical degrees of infinite sequences of $S$ obtained by sequentially composing elements of $S$ on the right and left.
15. Fri 5:00pm. 1154-11-221
Robert L. Benedetto
Isolation of postcritically finite parameters in $p$-adic dynamical moduli spaces.
Abstract. Fix a prime number $p$, and let $f_c(z)=f(c,z)\in\mathbb{C}_p[[c]](z)$ be a one-parameter analytic family of rational functions of degree $d\ge2$ for $c$ in some open disk $D$. Suppose that all $2d-2$ critical points of $f_c$ are also analytic functions of $c$. A parameter $c$ is postcritically finite, or PCF, if all of the critical points of $f_c$ have finite forward orbit under the iteration of $f_c$. Under mild conditions, including that all critical points lie in the Fatou set, we show that any proper subdisk of $D$ contains only finitely many PCF parameters, even though $D$ itself may contain infinitely many. In particular, all PCF parameters of $f_c(z)=z^d+c$ are isolated in $p$-adic dynamical moduli space.
16. Fri 5:30pm. 1154-11-2083
Michelle Manes
Post-critically finite cubic polynomials. Preliminary report.
Abstract. We find all cubic post-critically finite (PCF) polynomials defined over $\mathbb Q$, up to conjugacy over $\operatorname{PGL}_2(\overline{\mathbb Q})$. The techniques involve finding normal forms for cubic polynomials that respect field of definition, adapting some techniques of Ingram on coefficient bounds, and a bit of Sage computation. The same ingredients allow us to tackle questions of potential good reduction for these functions.

SATURDAY MORNING

17. Sat 8:00am. 1154-11-395
Andrea Ferraguti
The inverse problem for arboreal Galois representations of index two.
Abstract. Let $F$ be a field of characteristic not 2 and $f$ be a monic and quadratic polynomial with coefficients in $F$. Let $\Omega_\infty$ be the automorphism group of the infinite tree associated to $f$, and let $M\le\Omega_\infty$ be a given maximal subgroup. In this talk, I will first explain how to produce necessary and sufficient conditions, depending exclusively on the post-critical orbit of $f$, for the arboreal representation $\rho_f$ pto have image equal to $M$. Our way of thinking allows to quickly recover classical results, such as Stoll's criterion for surjectivity and infinite index image for PCF quadratic polynomial, from a structural point of view that does not involve the use of ramification theory. Next, I will provide infinite families of examples for polynomials of the form $x^2+a$ over the rationals. Finally, I will briefly show how to prove that any two closed subgroups of $\Omega_\infty$ of index at most 2 are non-isomorphic as topological groups. This involves the use of a new invariant for topological groups endowed with a system of topological generators named graph of commutativity.
18. Sat 8:30am. 1154-14-831
John Lesieutre
Higher arithmetic degrees of rational maps. Preliminary report.
Abstract. Suppose that $f:X\dashrightarrow X$ is a dominant rational self-map of a variety defined over a number field. For a point $P$ on $X$, Kawaguchi and Silverman have defined the arithmetic degree of $f$ at $P$, a measure of the asymptotic growth rate of the heights of points $f^n(P)$. In this talk, I will introduce a definition of higher arithmetic degrees, measuring the growth rates of heights of higher-dimensional cycles. I will then describe efforts to develop a theory of arithmetic degrees in parallel to the much better established theory of dynamical degrees. This project is joint work with Nguyen-Bac Dang, Dragos Ghioca, Fei Hu, and Matthew Satriano.
19. Sat 9:00am. 1154-11-123
Matthew Satriano
New types of heights with connections to the Batyrev-Manin and Malle Conjectures.
Abstract. The Batyrev-Manin conjecture gives a prediction for the asymptotic growth rate of rational points on varieties over number fields when we order the points by height. The Malle conjecture predicts the asymptotic growth rate for number fields of degree d when they are ordered by discriminant. The two conjectures have the same form and it is natural to ask if they are, in fact, one and the same. We develop a theory heights on stacks, and give a conjecture for the growth rate of points on stacks which specializes to the two aforementioned conjectures.
20. Sat 9:30am. 1154-14-1622
Yohsuke Matsuzawa
On Kawaguchi-Silverman conjecture for endomorphisms of rationally connected varieties.
Abstract. The Kawaguchi-Silverman conjecture asserts that the arithmetic degrees of Zariski dense orbits of a rational self-map are equal to the dynamical degree of the map. I will talk about the proof of the conjecture for endomorphisms of smooth projective rationally connected varieties admitting an int-amplified endomorphism. This is a joint work with Shou Yoshikawa.
21. Sat 10:00am. 1154-37-1621
Benjamin Hutz
Automorphism loci for endomorphisms of $\mathbb P^1$. Preliminary report.
Abstract. I will present some recent classification results for endomorphisms of $\mathbb P^1$ with nontrivial automorphisms stemming from the groups I worked with at the summer REU program at the Institute for Computational and Experimental Research in Mathematics (ICERM) in the summer of 2019. In characteristic 0, these results concern classifying families of maps of degree 2–4 with non-trivial automorphisms. In characteristic $p>0$, these results concern the realizability of finite subgroups of $\operatorname{PGL}_2(\mathbb F_p)$ as automorphism groups. The focus will be open questions stemming from these projects.
22. Sat 10:30am. 1154-11-1924
Jamie Juul
Arboreal Galois representations of PCF quadratic polynomials. Preliminary report.
Abstract. Let $K$ be a number field and $f(x)\in K[x]$ be a quadratic post-critically finite polynomial. We study the Galois groups $\operatorname{Gal}(K(f^{-n}(t))/K(t))$. The case $t$ is transcendental over $K$ was studied in previous unpublished work of Pink; we give new proofs of his results. We also examine the case $t\in K$. This is joint work with Rob Benedetto, Dragos Ghioca, and Tom Tucker.
23. Sat 11:00am. 1154-37-818
David Krumm
Algebraic preperiodic points of entire transcendental functions. Preliminary report.
Abstract. Motivated by questions in transcendental number theory, K. Mahler asked in 1976 whether there exists an entire tran- scendental function $f:\mathbb{C}\to\mathbb{C}$ with the property that $f(\overline{\mathbb Q})\subset\overline{\mathbb Q}$ and $f^{-1}(\overline{\mathbb Q})\subset\overline{\mathbb Q}$. Mahler's question was answered in the affirmative by Marques and Moreira in 2016. In this talk we will discuss some dynamical properties of this type of function $f$, in particular the structure of the directed graph of algebraic preperiodic points of $f$.
24. Sat 11:30am. 1154-37-668
John Doyle
Moduli spaces for dynamical systems with level structure.
Abstract. We construct moduli spaces for endomorphisms of projective space together with a dynamical notion of level structure — namely, marked points satisfying specified orbit relations. We will discuss various properties of these spaces, and we will mention a result that underscores a connection with the dynamical uniform boundedness conjecture of Morton and Silverman.