Algebraic, Complex and Arithmetic Dynamics (2022)

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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.