Algebraic, Complex and Arithmetic Dynamics (2022)

Date & Time

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.

  • Agendaplus--large


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


    10:00 - 11:00 AMRomain Dujardin | On the Dynamics of Uniformly Hyperbolic Hénon Maps
    11:30 - 12:30 PMJasmin Raissy | Spiralling Domains in Dimension 2
    5:00 - 6:00 PMPierre Berger | Analytic Pseudo-Rotations
    6:15 - 7:15 PMMartin Möller | Abelian Differentials: Geometry and Notions of Special Subvarieties


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


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


    10:00 - 11:00 AMRoland Roeder | Dynamics of Groups of Automorphisms of Character Varieties and Fatou/Julia Decomposition for Painlevé 6
    11:30 - 12:30 PMMisha Hlushchanka | Rational Maps, Julia Sets and Iterated Monodromy Groups: Complexity and Decomposition
    5:00 - 6:00 PMOpen Discussion
    6:15 - 7:15 PMOpen Discussion
  • Abstracts & Slidesplus--large

    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.

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