Algebraic, Complex and Arithmetic Dynamics

Date & Time

Laura DeMarco, Northwestern University
Mattias Jonsson, University of Michigan

In this symposium series, we aim to formulate a vision for future developments in complex, algebraic and arithmetic dynamics. The theme of this first symposium was Heights and Complexity in Algebraic Dynamics. We discussed arithmetic and algebraic features of dynamics on algebraic varieties. Important questions relate to the values and computations of dynamical degrees and heights; intersection (and unlikely intersection) problems in families of maps; and uniform bounds on dynamically  distinguished subvarieties, independent of the map. The specific goals of the week were to (1) pose questions, (2) determine where we are today with the already-existing big questions in the field, and (3) present the guiding principles and new methods that underlie our approaches to these problems. There were 12 research talks, one overview talk, two problem sessions and a reference collection session.

  • Reportplus--large

    Research talks

    Jason Bell gave a talk with the title “Some dynamical problems motivated by questions in noncommutative algebra,” illustrating that problems in algebraic dynamics sometimes arise from unexpected areas of mathematics. Results about abelian varieties have given rise to many interesting conjectures in arithmetic and algebraic dynamics. Charles Favre explained joint work with Thomas Gauthier on a version of the dynamical André-Oort conjecture, as formulated by Baker and DeMarco. The title was “Heights of polynomial dynamical pairs.” K3 surfaces play an important role in both geometry and dynamics. In his talk “Dynamics on K3s: Berkovich and tropical versions,” Simion Filip explained a non-Archimedean versions of results on the complex dynamics on K3 surfaces. Holly Krieger explained joint work with DeMarco and Ye, in which they use the Arakelov-Zhang height pairing and quantitative equidistribution results to give uniform bounds on the number of common preperiodic points for certain families of rational maps. The title was “The Arakelov-Zhang pairing and dynamical heights.” In his talk “Numerical dimension revisited,” John Lesieutre explained how dynamics can be used to construct divisors with weak positivity properties that in fact give counterexamples to conjectures (and theorems) in birational geometry. Nicole Looper gave a talk, “Uniform boundedness, equidistribution and the arithmetic of dynamically small points,” explaining how celebrated conjectures in Diophantine geometry implies conjectures in arithmetic dynamics, such as the uniform boundedness conjecture by Morton and Silverman for unicritical polynomials. The Kawaguchi-Silverman conjecture predicts a relation between the growth of degrees and heights under iteration. In his talk “Some topics from Kawaguchi-Silverman conjecture,” Yohsuke Matsuzawa described recent progress. Curt McMullen gave a talk with the title “Billiards and the arithmetic of nonarithmetic groups,” in which he presented work on the Hecke groups G_n, with particular emphasis on the (nonarithmetic) group G_5, the matrices this contains, and its connection to billiards.

    Keiji Oguiso explained work analyzing whether or not the automorphism group of a smooth projective variety is finitely generated. The title was “Inertia groups, decomposition groups and smooth projective varieties with nonfinite generated automorphism groups.” In his talk, “The Betti foliation, the canonical height and the geometric Bogomolov conjecture,” Junyi Xie explained recent joint work with Cantat, Gao and Habegger, in which they prove the geometric Bogomolov conjecture over (possibly higher-dimensional) function fields of characteristic zero. The equivariant Minimal Model Program aims to classify (weakly) polarized endomorphisms of projective varieties. De-Qi Zhang gave a talk titled “Equivariant Minimal Model Program, with a view toward algebraic and arithmetic dynamics” in which he discussed recent progress, together with some applications. The height pairing on curves introduced by Shouwu Zhang has played a crucial role in arithmetic dynamics and other fields. In his talk, “Admissible pairing of algebraic cycles,” Zhang proposed a higher-dimensional generalization, together with dynamical applications.

    Other talks and activities

    Joe Silverman gave an overview of what is known about the growth of heights and degrees under iteration of a rational self-map of a normal projective variety defined over a number field. This talk served as natural starting point for the talks of Matsuzawa and D.Q. Zhang.

    We organized two problem sessions. A wide range of open problems and possible new directions were proposed at these sessions. We will type up notes and post them to the organizers’ webpage.

    In order to take stock of the current state of knowledge and to help researchers in the future, we organized a session where the participants collected references, detailing progress on a number of important problems and conjectures, such as dynamical degree computations, the dynamical Manin-Mumford problem or the dynamical Mordell-Lang problem. We assigned a group of participants to each problem and asked them to write down a list of key references with a very brief description of the content for each one.

    Finally, the format of the symposium gave ample opportunities for informal discussions, explanations of technical points as well as philosophical ideas. These kinds of interactions often form the embryo of future research, and we are confident that many new results and collaboration will result from this first meeting in the symposium series.

  • Agenda & Slides/Notesplus--large


    10:00 - 11:00 AMCharles Favre | Heights of Polynomial Dynamical Pairs
    View Notes (PDF)
    11:30 - 12:30 PMJoseph Silverman | Dynamical Degrees and Arithmetic Degrees: History, Conjectures and Future Directions
    View Slides (PDF)
    5:00 - 6:00 PMHolly Krieger | The Arakelov-Zhang Pairing and Dynamical Heights
    View Notes (PDF)
    6:15 - 7:15 PMYohsuke Matsuzawa | Some Topics from Kawaguchi-Silverman Conjecture
    View Notes (PDF)


    10:00 - 11:00 AMCurtis McMullen | Billiards and the Arithmetic of Non-Arithmetic Groups
    View Slides (PDF)
    11:30 - 12:30 PMSimion Filip | Dynamics on K3s: Berkovich and Tropical Versions
    View Notes (PDF)
    5:00 - 6:00 PMJunyi Xie | The Betti Foliation, the Canonical Height and the Geometric Bogomolov Conjecture
    View Notes (PDF)
    6:15 - 7:15 PMProblem Session


    10:00 - 11:00 AMShou-Wu Zhang | Admissible Pairing of Algebraic Cycles
    View Notes (PDF)
    11:30 - 12:30 PMJason Bell | Some Dynamical Problems Motivated by Questions in Noncommutative Algebra
    View Notes (PDF)
    5:00 - 6:00 PMJohn Lesieutre | Numerical Dimension Revisited
    View Notes (PDF)
    6:15 - 7:15 PMNicole Looper | Uniform Boundedness, Equidistribution and the Arithmetic of Dynamically Small Points
    View Slides (PDF)
    8:30 - 9:30 PMConcert: Messiaen: Quatuor pour la fin du temps


    9:45 - 2:00 PMGuided Hike to Partnach Gorge
    5:00 - 6:00 PMAd Hoc Talk
    6:15 - 7:15 PMReference Collection


    10:00 - 11:00 AMDe-Qi Zhang | Equivariant Minimal Model Program with a View Towards Algebraic and Arithmetic Dynamics
    View Slides (PDF)
    11:30 - 12:30 PMKeiji Oguiso | Inertia Groups, Decomposition Groups and Smooth Projective Varieties with Nonfinite Generated Automorphism Groups
    View Notes (PDF)
    5:00 - 6:00 PMProblem Session
    6:15 - 7:15 PMMatt Baker | On Some Unlikely Coincidences
  • Abstractsplus--large

    Jason Bell
    University of Waterloo

    Some Dynamical Problems Motivated by Questions in Noncommutative Algebra

    Bell will give an overview of some of the dynamical questions that arise when studying the representation theory of algebras in noncommutative projective geometry, highlighting some of the results already obtained in this direction.


    Charles Favre
    École Polytechnique

    Heights of Polynomial Dynamical Pairs

    Favre will discuss some of our progress with Thomas Gauthier in the problem of unlikely intersection in polynomial dynamics (over a one-dimensional base defined over a number field). Our results lead to further insights into the dynamical Andre-Oort conjecture.


    Simon Filip
    Institute for Advanced Study & Clay Mathematics Institute

    Dynamics on K3s: Berkovich and Tropical Versions

    Filip will start by recalling the basic facts about algebraic automorphisms of K3 surfaces. He will then explain how to extend some of the results to the non-archimedean setting and discuss the resulting dynamical systems. The interesting part of the dynamics has an explicit, elementary description in terms of tropical geometry. Filip will end with some questions and discuss possible applications.


    Holly Krieger
    University of Cambridge

    The Arakelov-Zhang Pairing and Dynamical Heights

    For any two rational maps of the Riemann sphere with algebraic coefficients, the Arakelov-Zhang pairing of their canonical heights provides an arithmetic measure of the dynamical distance between the two maps. Krieger will discuss how this pairing can be used, together with quantitative equidistribution, to provide bounds on points of small height for both maps, as done in recent joint work with DeMarco and Ye. Krieger will highlight some of the many open questions about the behavior of this pairing in moduli.


    John Lesieutre
    Pennsylvania State University

    Numerical Dimension Revisited

    The Iitaka dimension of a line bundle \(D\) on a projective variety \(X\) is the dimension of the image of the rational map given by \(|mD|\) for large and divisible \(m\). The Iitaka dimension is not a numerical invariant of \(D\), and there are several approaches to constructing a “numerical dimension,” which should be an analogous invariant depending only on the numerical class of \(D\). Lesieutre will discuss some divisors of dynamical origin whose behavior with respect to these invariants is pathological and which provide counterexamples to some conjectures from birational geometry. The examples hinge on a sort of dynamical positivity property, which also arises in arithmetic contexts. Lesieutre will then pose some related problems about degree growth on varieties with large groups of pseudoautomorphisms.


    Nicole Looper
    University of Cambridge

    Uniform Boundedness, Equidistribution and the Arithmetic of Dynamically Small Points

    In this talk, Looper will discuss a uniform boundedness theorem for unicritical, along with the relevant tools from, Diophantine geometry. Looper will also discuss connections to other results concerning points of small canonical height relative to polynomials.


    Yohsuke Matsuzawa
    University of Tokyo

    Some Topics from Kawaguchi-Silverman Conjecture

    Matsuzawa will talk about some topics from Kawaguchi-Silverman conjecture, which asserts that arithmetic degrees of Zariski dense orbits under self-rational maps are equal to the dynamical degree of the map. The situation is completely different whether the self-rational map is a self-morphism of a projective variety or not. Tools from birational geometry are very helpful for self-morphisms, but it seems these are not sufficient to understand the arithmetic of self-rational maps. We have to know the behavior of the height function associated with the indeterminacy locus of the self-map. Matsuzawa will discuss recent progress on the conjecture for self-morphisms of projective varieties, and also mention algebraically stable self-rational maps.


    Curtis McMullen
    Harvard University

    Billiards and the Arithmetic of Non-Arithmetic Groups

    The classical Hecke groups \(G_n\) in \(SL_2(R)\), also known as the (2,n,infinity) triangle groups, are non-arithmetic for most n: no simply criterion is known for describing the matrices they contain. McMullen will discuss new insights into these groups arising from their connection with billiards in polygons and totally geodesic curves in moduli space.


    Keiji Oguiso
    University of Tokyo

    Inertia Groups, Decomposition Groups and Smooth Projective Varieties with Nonfinite Generated Automorphism Groups

    The so-called ‘Coble problem’ concerning complexities of the decomposition group and inertia group of the special smooth rational curve on a classic complex Coble surface is a long-standing problem which is still open. In this talk, Oguiso will attempt to negatively answer another long-standing problem — the finite generation problem of the automorphism group of a smooth projective variety of any dimension \(\ge 2\) over an algebraically closed field, under the assumption that the base field is not an algebraic closure of a finite field or not of characteristic \(2\). In our construction, the decomposition group of a smooth rational curve of some special K3 surface, which is closely related to classical Coble surfaces, and the arithmetic of the base field play essential and delicate roles, as Oguiso will explain. It also turns out that the inertia group in our construction has rich complex dynamics. This talk is partly based on Oguiso’s joint work with Coung-Tien Dinh and is also much inspired by the works of Lesieutre and Dolgachev.


    Joseph Silverman
    Brown University

    Dynamical Degrees and Arithmetic Degrees: History, Conjectures and Future Directions

    Silverman will give an overview of various quantities that are used to measure the complexity of an algebraic dynamical system \(f: X\to X\), including the dynamical degree \(\delta(f)\), which gives a coarse measure of the geometric complexity of the iterates of \(f\), the arithmetic degree \(\alpha(f,P)\), which gives a coarse measure of the arithmetic complexity of the orbit of \(P\in{X}(\overline{\mathbb{Q}})\), and various versions of the canonical height \(\hat{h}_f(P)\) that provide more refined measures of arithmetic complexity. Emphasis will be placed on open problems and directions for exploration.


    Junyi Xie
    Université de Rennes

    The Betti Foliation, the Canonical Height and the Geometric Bogomolov Conjecture

    With Cantat, Habegger and Gao, Xie will prove the geometric Bogomolov conjecture over a function field of characteristic zero. This generalizes recent work of Habegger and Gao, who proved the geometric Bogomolov conjecture over a function field of a curve of characteristic zero.


    De-Qi Zhang
    National University of Singapore

    Equivariant Minimal Model Program with a View Toward Algebraic and Arithmetic Dynamics

    Zhang will elaborate the notion of ‘int-amplified’ endomorphism f of a normal projective variety X, a property weaker than ‘polarized’ yet preserved by products. Zhang will show that the existence of such a single f guarantees that every Minimal Model Program (MMP) is equivariant w.r.t. a finite-index submonoid of the whole monoid SEnd(X) of all surjective endomorphisms of X. Applications of the equivariant MMP are discussed: Kawaguchi-Silverman conjecture on the equivalence of arithmetic and dynamic degrees, and characterization of a subvariety with Zariski dense periodic points. Some parts are based on joint work with Cascini and Meng.


    Shou-Wu Zhang
    Princeton University

    Admissible Pairing of Algebraic Cycles

    For a smooth and projective variety X over a global field of dimension n with an adelic polarization, Zhang proposes canonical local and global height pairings for two cycles Y , Z of pure codimension p, q satisfying p + q = n + 1. Zhang will discuss some applications to algebraic dynamical systems and Shimura varieties.

  • Participantsplus--large
    Matthew BakerGeorgia Institute of Technology
    Jason Bell University of Waterloo
    Nguyen-Bac DangStony Brook University
    Laura DeMarcoNorthwestern University
    Jeff DillerUniversity of Notre Dame
    Charles FavreÉcole Polytechnique
    Simion FilipInstitute for Advanced Study and Clay Mathematics Institute
    Patrick IngramYork University
    Mattias JonssonUniversity of Michigan
    Shu KawaguchiDoshisha University
    Holly KriegerUniversity of Cambridge
    John LesieutrePennsylvania State University
    Nicole LooperUniversity of Cambridge
    Yohsuke MatsuzawaUniversity of Tokyo
    Myrto MavrakiNorthwestern University
    Curtis McMullenHarvard University
    Keiji OguisoUniversity of Tokyo
    Matthew SatrianoUniversity of Waterloo
    Joseph SilvermanBrown University
    Tom TuckerUniversity of Rochester
    Junyi XieUniversité de Rennes
    Shou Wu ZhangPrinceton University
    De-Qi ZhangNational University of Singapore
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