# Algebraic, Complex and Arithmetic Dynamics (2022)

Date & Time

Organizers:
Laura DeMarco, Harvard University
Mattias Jonsson, University of Michigan

The goal of Simons Symposium on Algebraic, Complex and Arithmetic Dynamics is to formulate a vision for future developments in complex, algebraic, and arithmetic dynamics, with a focus on methods coming from complex dynamics. Organizers aim to take stock of recent developments in complex dynamics, but also explore the important role that complex methods play in questions of more algebraic or arithmetic nature.

Discussion topics include:

• Recent developments in complex dynamics
• Critical orbit relations in complex and arithmetic dynamics
• The role of complex dynamics for questions of arithmetic nature

### Meeting Report

In this symposium series, we aim to formulate a vision for future developments in complex, algebraic and arithmetic dynamics. The first symposium was centered around questions with an arithmetic flavor, especially the concept of heights and notions of dynamical complexity. The theme of this second symposium was “Complex Dynamics and Algebraic Structures.” The lectures provided an overview of key research directions in complex algebraic dynamical systems, with an emphasis on the complex-analytic theory. There were 14 lectures and 2 open problem sessions. The speakers were asked to focus on questions for future research, while highlighting significant developments in the field from recent years.

#### Research talks

The talks covered a wide range of topics in complex and algebraic dynamics. Each lecturer was asked to address what might come next in their research area, while describing important recent developments in the field.

Beginning with questions in one-dimensional dynamics, Mikhail Lyubich provided an overview of recent developments in anti-holomorphic dynamics and reflection groups. Mikhail Hlushchanka presented recent results about the topology of Julia sets and canonical decompositions of rational maps on the Riemann sphere. Nguyen-Bac Dang addressed iterated monodromy groups for polynomials in dimension one, while utilizing a higher-dimensional dynamical system to study the spectrum of the graph Laplacian. In a related direction, Han Peters presented questions about graphs with a physical motivation, leading back to questions about related one-dimensional dynamical systems and bifurcations. Curtis McMullen provided an overview of results and questions about both one- and higher-dimensional algebraic dynamical systems, tying in the theory of billiard dynamics and the classification of Teichmüller curves.

Related to this, Martin Möller described results about compactifying spaces of abelian diﬀerentials, describing questions about the arithmetic of these moduli spaces. Also addressing questions about the arithmetic of moduli spaces, Niki Myrto Mavraki presented recent results about families of abelian varieties and explained their relevance to studying more general families of complex algebraic dynamical systems. Similarly, Thomas Gauthier emphasized arithmetic-dynamical methods, raising interesting new questions about heights and bifurcation measures. On the topic of bifurcations, Fabrizio Bianchi described the current status of bifurcation theory in dimensions one and higher, with a presentation on the bifurcation currents and measures and what remains unresolved. We also had several talks presenting the state of the art of complex dynamics in higher dimensions in a number of diﬀerent settings. Jasmin Raissy discussed the local theory and recent developments in our study of holomorphic germs tangent to the identity transformation, with global implications for endomorphisms of projective spaces.

Roland Roeder presented questions about a particular family of automorphisms of aﬃne surfaces, related to the Painlevé 6 equation. Romain Dujardin discussed polynomial automorphisms of the aﬃne plane, especially the hyperbolic Hénon maps and questions about their Julia sets. Pierre Berger drew parallels between Hénon maps and analytic maps on cylinders, relating the analysis of Julia sets to classical questions about symplectomorphisms. And finally, Serge Cantat provided a survey of results and questions about the dynamics of automorphisms of algebraic surfaces.

#### Other talks and activities

We organized two problem sessions. A wide range of open problems and possible new directions were proposed at these sessions. Not all invited participants were able to give a full-length talk, and the problem sessions made it possible for them to share some ideas and thoughts at the board. We will post notes to the organizers’ webpage.

The format of the symposium provided ample opportunities for informal discussions, explanations of technical points and deliberations on future research directions. We are confident that many new results and collaborations will result from this first meeting in the symposium series.

• Agenda

#### MONDAY, AUGUST 22

 10:00 - 11:00 AM Mikhail Lyubich | Conformal Dynamics: Julia Sets, Kleinian Groups, Schwarz Reflections and Algebraic Correspondences 11:30 - 12:30 PM Serge Cantat | Automorphisms of Complex Projective Surfaces: A Few Open Problems 5:00 - 6:00 PM Myrto Mavraki | Preperiodic Points in Families of Rational Maps 6:15 - 7:15 PM Curtis McMullen | Algebraic Curves and Complex Dynamics

#### TUESDAY AUGUST 23

 10:00 - 11:00 AM Romain Dujardin | On the Dynamics of Uniformly Hyperbolic Hénon Maps 11:30 - 12:30 PM Jasmin Raissy | Spiralling Domains in Dimension 2 5:00 - 6:00 PM Pierre Berger | Analytic Pseudo-Rotations 6:15 - 7:15 PM Martin Möller | Abelian Differentials: Geometry and Notions of Special Subvarieties

#### WEDNESDAY AUGUST 24

 3:30 - 4:30 PM Open Discussion 5:00 - 6:00 PM Han Peters | Graph Theory, Computational Complexity, Statistical Physics and Complex Dynamics

#### THURSDAY AUGUST 25

 10:00 - 11:00 AM Fabrizio Bianchi | Bifurcations: From One to Several Complex Variables 11:30 - 12:30 PM Bac Dang | Spectrum of the Laplacian on the Basilica Group and Holomorphic Dynamics 5:00 - 6:00 PM Thomas Gauthier | A Complex Analytic Approach to Sparsity, Rigidity and Uniformity in Arithmetic Dynamics 6:15 - 7:15 PM Problem Session

#### FRIDAY AUGUST 26

 10:00 - 11:00 AM Roland Roeder | Dynamics of Groups of Automorphisms of Character Varieties and Fatou/Julia Decomposition for Painlevé 6 11:30 - 12:30 PM Misha Hlushchanka | Rational Maps, Julia Sets and Iterated Monodromy Groups: Complexity and Decomposition 5:00 - 6:00 PM Open Discussion 6:15 - 7:15 PM Open Discussion
• Abstracts & Slides

Pierre Berger
CNRS-Sorbonne Université

Analytic Pseudo-Rotations

Pierre Berger will construct analytic symplectomorphisms of the cylinder or the sphere with zero or exactly two periodic points and which are not conjugated to a rotation. In the case of the cylinder, Berger will show that these symplectomorphisms can be chosen ergodic or to the opposite with local emergence of maximal order. This disproves a conjecture of Birkhoff (1941) and solves a problem of Herman (1998). We will note a connection with the J=J* problem.

Fabrizio Bianchi
CNRS – Université de Lille

Bifurcations: From One to Several Complex Variables
View Slides (PDF)

Fabrizio Bianchi will discuss the stability of holomorphic dynamical systems under perturbation. In dimension 1, the theory is now classical and is based on works by Lyubich, Mané-Sad-Sullivan, and DeMarco. Bianchi will review this theory and present a recent generalisation valid for families of endomorphisms in any dimension. Since classical 1-dimensional techniques no longer apply in higher dimensions, the approach is based on ergodic and pluripotential methods. Bianchi will list several open questions, as well as some partial results in these directions. This talk is partially based on joint works with M. Astorg, F. Berteloot, T.-C. Dinh, C. Dupont, Y. Okuyama, K. Rakhimov, and J. Taflin.

Serge Cantat
CNRS – University Rennes

Automorphisms of Complex Projective Surfaces: A Few Open Problems

Serge Cantat will describe a few open problems concerning the complex dynamics of (groups of) automorphisms and birational transformations of projective surfaces. The emphasis will be on topological dynamics, rather than stochastic properties.

Bac Dang
Université Paris Saclay

Spectrum of the Laplacian on the Basilica Group and Holomorphic Dynamics
View Slides (PDF)
View Slides (PDF)

In this talk, based on an ongoing work with Eric Bedford, Rostislav Grigorchuk and Mikhail Lyubich, Bac Dang will present how the spectrum of the Laplacian on the Basilica Schreier graphs is related to the iteration of the rational map and to the statistical behavior of the pullback of a particular line.

Romain Dujardin
Sorbonne Université

On the Dynamics of Uniformly Hyperbolic Hénon Maps

Romain Dujardin will study the dynamics of polynomial automorphisms of C^2 which are uniformly hyperbolic on their Julia sets. It may be surprising that many basic questions about the topology of the Julia set remain unanswered. In a joint work with Misha Lyubich, Dujardin will focus on mappings with disconnected Julia set and give a (tentative) classification of its connected components, which is reminiscent of classical one-variable polynomials.

Thomas Gauthier
Université Paris-Saclay

A Complex Analytic Approach to Sparsity, Rigidity and Uniformity in Arithmetic Dynamics

This talk is concerned with connections between arithmetic dynamics and complex dynamics. The first aim of the talk is to discuss several open problems from arithmetic dynamics and to explain how these problems are related to complex dynamical tool: bifurcation measures. If time allows, Thomas Gauthier will give a strategy to tackle several of those problems at the same time. This is based on a joint work in progress with Gabriel Vigny.

Misha Hlushchanka
Utrecht University

Rational Maps, Julia Sets and Iterated Monodromy Groups: Complexity and Decomposition
View Slides (PDF)

Sullivan’s dictionary and the theory of iterated monodromy groups bridge complex dynamics and geometric group theory. Misha Hlushchanka will discuss known and prospective relations between different measures of complexity of dynamical systems, fractal sets, and groups in these contexts.

Mikhail Lyubich
Stony Brook University

Conformal Dynamics: Julia Sets, Kleinian Groups, Schwarz Reflections and Algebraic Correspondences
View Slides (PDF)

Mikhail Lyubich will discuss the interplay between four branches of Conformal Dynamics: iterations of (anti-)rational maps, actions of Kleinian groups, dynamics generated by Schwarz reflections in quadrature domains, and algebraic correspondences. Lyubich will show examples of Schwarz reflections obtained by matings between anti-quadratic maps and the triangle modular group, and examples of Julia realizations for Apollonian-like gaskets. Some of these examples can be turned into others by means of a David surgery (e.g., the Apollonian Julia set to the Apollonian Kleinian group). The Schwarz reflection parameter space can be sometimes related to the parameter space of the Tricorn or of an appropriate anti-rational parabolic family. The latter can even be done by means of a quasiconformal straightening (making use of a classical Warschawski Theorem). For instance, this is the case for the Schwarz families obtained by univalent restrictions of Belyi-Shabat polynomials to appropriate disks. The associated algebraic correspondences are genereted by the deck transformations of these polynomials and by the reflections in the corresponding circles.

Myrto Mavraki
Harvard University

Preperiodic Points in Families of Rational Maps
View Slides (PDF)

In recent breakthroughs Dimitrov, Gao, Habegger and Kühne have established that the Bogomolov conjecture in arithmetic geometry holds uniformly across curves of a given genus at least 2. Inspired by the analogy between torsion points in abelian varieties and preperiodic points in a dynamical system, Zhang has proposed a dynamical analog of the Bogomolov conjecture. For instance, when can two rational maps share infinitely many common preperiodic points? Various authors have combined arithmetic and complex analytic ingredients to answer such questions. In this talk we discuss progress towards uniform versions of the dynamical Bogomolov conjecture for split rational maps. Our approach goes through a ‘relative’ Bogomolov problem in families of maps, analogous to the classical relative Bogomolov conjecture. Though related questions have been considered in the dynamical setting by Baker-DeMarco and Favre-Gauthier, many problems remain unsolved. The talk will feature works with Harry Schmidt and with Laura DeMarco.

Curtis McMullen
Harvard University

Algebraic Curves and Complex Dynamics
View Slides (PDF)

Curtis McMullen will survey some advances and open problems in complex dynamics, taking as the point of departure the study of curves in moduli space and algebraic
families of dynamical systems.

Martin Möller
Goethe Universität Frankfurt/Main

Abelian Differentials: Geometry and Notions of Special Subvarieties
View Slides (PDF)

Martin Möller will give a summary of the known (algebraic) geometry properties of the moduli spaces of abelian differentials, comparing them to the moduli spaces of curves. Moreover, Möller summarize known results about affine invariant subvarieties and speculate about notions of special points.

Han Peters
University of Amsterdam

Graph Theory, Computational Complexity, Statistical Physics and Complex Dynamics
View Slides (PDF)

Partition functions on graphs that are in some sense recursively defined often naturally leads to holomorphic dynamical systems. An elegant example was given in recent work of Ombra and Riveira-Letelier, where they study the partition function of the hard-core model on Cayley trees. This setting induces the iteration of a one-parameter family of rational functions. In fact, the zero locus of the partition functions coincides with the bifurcation locus of the associated family of rational maps. Therefore, it was shown that there exists a single-phase transition of infinite order.

In recent works with de Boer, Buys, Guerini and Regts, Han Peters demonstrated that this connection between the zero locus of partition functions and the bifurcation locus of associated rational functions persists in settings that have no clear dynamical interpretation. Examples are the family of all bounded degree graphs, for both the hard-core and the Ising model. Again, the zero locus coincides with (an interpretation of) the bifurcation locus. In current work the focus is on the setting that is most interesting from a physical perspective: graphs converging to a regular lattice. While there is no clear interpretation as a holomorphic dynamical system, both simulations and preliminary results demonstrate the potential of methods from complex dynamical systems in this setting.

Jasmin Raissy
Institut de Mathématiques de Bordeaux, Université de Bordeaux

Spiralling Domains in Dimension 2
View Slides (PDF)

In this talk, Jasmin Raissy will present a joint work in progress with Xavier Buff. Raissy will study the dynamics of polynomials endomorphisms of $$\mathbb{C}^2$$ which are tangent to the identity at a fixed point. The goal is to show the existence of such maps for which the immediate basin of attraction of the fixed point has an infinite number of distinct invariants connected components, where the orbits converge to the fixed point without being tangent to any direction.

Roland Roeder
Indiana University Purdue University Indianapolis

Dynamics of Groups of Automorphisms of Character Varieties and Fatou/Julia Decomposition for Painlevé 6
View Slides (PDF)

Roland Roeder will study the dynamics of the group of holomorphic automorphisms of the affine cubic surfaces
$$S_{A,B,C,D} = {(x,y,z) \in C^3 : x^2 + y^2 + z^2 + xyz = Ax + By+Cz+D}$$
where A,B,C, and D are complex parameters. It arises naturally in the dynamics on character varieties and it also describes the monodromy of the famous Painleve 6 differential equation. Roeder will explore the Fatou/Julia dichotomy for this group action (defined analogously to the usual definition for iteration of a single rational map) and also the Locally Discrete / Non-Discrete dichotomy (a non-linear version from the classical discrete/non-discrete dichotomy for Lie groups). The interplay between these two dichotomies proves several results about the topological dynamics of this group. Moreover, Roeder will show the coexistence of non-empty Fatou sets and Julia sets with non-trivial interior for a large set of parameters. Several open questions related to our work will be described. This is joint work with Julio Rebelo.

Subscribe to MPS announcements and other foundation updates