Algebraic, Complex and Arithmetic Dynamics
Organizers:
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 alreadyexisting 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.

Report
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 AndréOort 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 nonArchimedean versions of results on the complex dynamics on K3 surfaces. Holly Krieger explained joint work with DeMarco and Ye, in which they use the ArakelovZhang 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 ArakelovZhang 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 KawaguchiSilverman conjecture predicts a relation between the growth of degrees and heights under iteration. In his talk “Some topics from KawaguchiSilverman 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 higherdimensional) function fields of characteristic zero. The equivariant Minimal Model Program aims to classify (weakly) polarized endomorphisms of projective varieties. DeQi 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 higherdimensional 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 selfmap 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 ManinMumford problem or the dynamical MordellLang 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/Notes
MONDAY
10:00  11:00 AM Charles Favre  Heights of Polynomial Dynamical Pairs
View Notes (PDF)11:30  12:30 PM Joseph Silverman  Dynamical Degrees and Arithmetic Degrees: History, Conjectures and Future Directions
View Slides (PDF)5:00  6:00 PM Holly Krieger  The ArakelovZhang Pairing and Dynamical Heights
View Notes (PDF)6:15  7:15 PM Yohsuke Matsuzawa  Some Topics from KawaguchiSilverman Conjecture
View Notes (PDF)TUESDAY
10:00  11:00 AM Curtis McMullen  Billiards and the Arithmetic of NonArithmetic Groups
View Slides (PDF)11:30  12:30 PM Simion Filip  Dynamics on K3s: Berkovich and Tropical Versions
View Notes (PDF)5:00  6:00 PM Junyi Xie  The Betti Foliation, the Canonical Height and the Geometric Bogomolov Conjecture
View Notes (PDF)6:15  7:15 PM Problem Session WEDNESDAY
10:00  11:00 AM ShouWu Zhang  Admissible Pairing of Algebraic Cycles
View Notes (PDF)11:30  12:30 PM Jason Bell  Some Dynamical Problems Motivated by Questions in Noncommutative Algebra
View Notes (PDF)5:00  6:00 PM John Lesieutre  Numerical Dimension Revisited
View Notes (PDF)6:15  7:15 PM Nicole Looper  Uniform Boundedness, Equidistribution and the Arithmetic of Dynamically Small Points
View Slides (PDF)8:30  9:30 PM Concert: Messiaen: Quatuor pour la fin du temps THURSDAY
9:45  2:00 PM Guided Hike to Partnach Gorge 5:00  6:00 PM Ad Hoc Talk 6:15  7:15 PM Reference Collection FRIDAY
10:00  11:00 AM DeQi Zhang  Equivariant Minimal Model Program with a View Towards Algebraic and Arithmetic Dynamics
View Slides (PDF)11:30  12:30 PM Keiji Oguiso  Inertia Groups, Decomposition Groups and Smooth Projective Varieties with Nonfinite Generated Automorphism Groups
View Notes (PDF)5:00  6:00 PM Problem Session 6:15  7:15 PM Matt Baker  On Some Unlikely Coincidences 
Abstracts
Jason Bell
University of WaterlooSome 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 PolytechniqueHeights 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 onedimensional base defined over a number field). Our results lead to further insights into the dynamical AndreOort conjecture.
Simon Filip
Institute for Advanced Study & Clay Mathematics InstituteDynamics 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 nonarchimedean 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 CambridgeThe ArakelovZhang Pairing and Dynamical Heights
For any two rational maps of the Riemann sphere with algebraic coefficients, the ArakelovZhang 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 UniversityNumerical 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 CambridgeUniform 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 TokyoSome Topics from KawaguchiSilverman Conjecture
Matsuzawa will talk about some topics from KawaguchiSilverman conjecture, which asserts that arithmetic degrees of Zariski dense orbits under selfrational maps are equal to the dynamical degree of the map. The situation is completely different whether the selfrational map is a selfmorphism of a projective variety or not. Tools from birational geometry are very helpful for selfmorphisms, but it seems these are not sufficient to understand the arithmetic of selfrational maps. We have to know the behavior of the height function associated with the indeterminacy locus of the selfmap. Matsuzawa will discuss recent progress on the conjecture for selfmorphisms of projective varieties, and also mention algebraically stable selfrational maps.
Curtis McMullen
Harvard UniversityBilliards and the Arithmetic of NonArithmetic Groups
The classical Hecke groups \(G_n\) in \(SL_2(R)\), also known as the (2,n,infinity) triangle groups, are nonarithmetic 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 TokyoInertia Groups, Decomposition Groups and Smooth Projective Varieties with Nonfinite Generated Automorphism Groups
The socalled ‘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 longstanding problem which is still open. In this talk, Oguiso will attempt to negatively answer another longstanding 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 CoungTien Dinh and is also much inspired by the works of Lesieutre and Dolgachev.
Joseph Silverman
Brown UniversityDynamical 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 RennesThe 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.
DeQi Zhang
National University of SingaporeEquivariant Minimal Model Program with a View Toward Algebraic and Arithmetic Dynamics
Zhang will elaborate the notion of ‘intamplified’ 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 finiteindex submonoid of the whole monoid SEnd(X) of all surjective endomorphisms of X. Applications of the equivariant MMP are discussed: KawaguchiSilverman 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.
ShouWu Zhang
Princeton UniversityAdmissible 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.

Participants
Matthew Baker Georgia Institute of Technology Jason Bell University of Waterloo NguyenBac Dang Stony Brook University Laura DeMarco Northwestern University Jeff Diller University of Notre Dame Charles Favre École Polytechnique Simion Filip Institute for Advanced Study and Clay Mathematics Institute Patrick Ingram York University Mattias Jonsson University of Michigan Shu Kawaguchi Doshisha University Holly Krieger University of Cambridge John Lesieutre Pennsylvania State University Nicole Looper University of Cambridge Yohsuke Matsuzawa University of Tokyo Myrto Mavraki Northwestern University Curtis McMullen Harvard University Keiji Oguiso University of Tokyo Matthew Satriano University of Waterloo Joseph Silverman Brown University Tom Tucker University of Rochester Junyi Xie Université de Rennes Shou Wu Zhang Princeton University DeQi Zhang National University of Singapore