Algebraic, Complex, and Arithmetic Dynamics (2024)

*Participants may explore the hotel property and its surrounding areas as well as engage in informal discussion with other participants.

Jason Bell
University of Waterloo

p-Adic Interpolation

We give an overview of p-adic interpolation methods and highlight some of their applications in arithmetic dynamics.

Xavier Buff
Institut Mathématiques de Toulouse

Differential Calculus in Holomorphic Dynamics

We will present methods introduced by Adam Epstein to address transversality problems in holomorphic dynamics in complex dimension 1. One associates to each rational map various dynamically relevant vector spaces and linear maps, and then studies the kernel and/or the image of those linear maps. The key ingredient in the proof, which goes back to William Thuston, is the injectivity of the operator id – f_* acting on meromorphic integrable quadratic differentials. All statements are purely algebraic, but we are not aware of an algebraic proof of the key ingredient.

Slides (PDF)

Serge Cantat
CNRS

Stable Manifolds and Entire Curves

In complex dimension 2, stable manifolds of holomorphic diffeomorphisms are entire curves; that is, they are holomorphically parametrized by the affine complex line. Positive currents, Nevanlinna theory and Hodge theory can then be used to study these curves. In this talk, Serge Cantat will describe some of the results from this circle of ideas.

Laura DeMarco
Harvard University

Algebraic, Complex, and Arithmetic Dynamics

In this talk, Laura DeMarco will present a conjecture that grew out of discussions during the first two symposia. It includes as special cases many well-known theorems and challenging conjectures in the areas of algebraic, complex, and arithmetic dynamics. This is joint work with Myrto Mavraki.

Romain Dujardin
Sorbonne Université

Rigidity for Polynomial Automorphisms of C^2

In joint work with Serge Cantat, we establish several new rigidity results for automorphisms with positive entropy, addressing questions such as the smoothness of Julia sets, existence of invariant foliations or characterizations of real-analytic conjugacy classes. Most of these questions are related to properties of stable and unstable eigenvalues of saddle periodic points and the algebraic and geometric constraints they have to satisfy. Another pervasive tool is the existence and properties of the dynamical Green function.

Thomas Gauthier
Université Paris-Saclay

Potential-Theoretic Tools for Local Problems in Parameter Spaces of Endomorphisms

When studying the global (or local) geography of parameter spaces of endomorphism of projective spaces, one can think of using tools inherited from potential theory, such as closed positive currents. One of the main issues we then encounter is to be able to prove that those objects actually carry useful information. In this talk, Thomas Gauthier will present various situations where we can find this information in a parameter/dynamics similarity phenomenon.

Mattias Jonsson
University of Michigan

Dynamical Degenerations and Berkovich Spaces

Non-Archimedean analytic spaces in the sense of Berkovich can be used to study degeneration of various types of complex analytic objects. Mattias Jonsson will survey some known results about degenerations in complex dynamics, with an emphasis on the methods used in the proofs.

Sarah Koch
University of Michigan

Complex Dynamics without Dynamics

Sarah Koch will discuss a variety of dynamical results, many in the realm of PCF rational maps, that arise from exploiting “non-dynamical” ideas and constructions.

Slides (PDF)

Yusheng Luo
Cornell University

Degenerations of Hyperbolic Components

From a dynamics point of view, a degenerating sequence of conjugacy classes of rational maps in a hyperbolic component usually develops some wide rectangles (arc-degeneration) or annuli (loop-degeneration) in the dynamical plane. In this talk, we will discuss some methods and techniques to prevent these degenerations. Using a hyperbolic component of a Sierpiński carpet Julia set as an example, we will illustrate how renormalization methods can be used to bound the arc-degeneration and how the ideas from the Berkovich spaces can be used to prevent loop-degeneration. Furthermore, we will talk about some limitations, technical challenges and potential generalizations for these methods.

Yohsuke Matsuzawa
Osaka Metropolitan University

Growth of Local Height Functions Along Orbits

For a dominant self-map f on a projective variety, we explore the problem of whether the ratio (local height of f^n(x))/( ample height of f^n(x)) tends to zero as n approaches infinity. This problem was solved in dimension one by Silverman, with applications to the dynamical Lang-Siegel problem. In higher dimensions, there are two main obstacles: Diophantine approximation (specifically, the upper bound of local height functions) and the growth rate of ample height functions along orbits.

For the first obstacle, Yohsuke Matsuzawa will introduce how Roth’s theorem and Vojta’s conjecture can be applied. It appears to be very difficult to prove higher dimensional analogue of Silverman’s theorem in full generality without assuming Vojta’s conjecture. However, Matsuzawa will explain that a weaker statement involving Banach density can be proven unconditionally. Regarding the second obstacle, it is well-understood when f is a morphism (on a projective variety). When f is a rational map, the problem becomes much more challenging. Matsuzawa will explain that when f is a cohomologically hyperbolic map, the growth of the ample height is sufficiently well-understood to the extent that we can prove a sufficient condition for our problem. This is an application of recent work by Junyi Xie on recursive inequalities of pull-backs of divisors.

Myrto Mavraki
University of Toronto

Uniform Unlikely Intersections

Myrto Mavraki will describe techniques used to obtain uniform bounds in families of arithmetic dynamical systems

Curtis McMullen
Harvard University

Billiards, Heights and Hodge Theory

What are the slopes of periodic billiard trajectories in a regular polygon? We will connect this dynamical question and others to methods lying at the nexus of complex and arithmetic geometry. In particular, we will discuss a new height on P^1(K), determined by a combination of arithmetic and the Hodge norm on the cohomology of an Abelian variety with real multiplication by K. Along the way, we will encounter issues of chaos and decidability, first appearing in polygons with 7 and 12 sides.

Sildes (PDF)

Joseph Silverman
Brown University

Height Counting Functions

Joseph Silverman will start with a 30-year old conjecture that describes the coarse growth of height counting functions on algebraic varieties, show how it leads to a dynamical orbit propagation conjecture, and then use height counting functions to prove a case of the orbit propagation conjecture (joint work with Hector Pasten).

Valentino Tosatti
New York University

Two Applications of Semicontinuity to Complex Dynamics

Valentino Tosatti will discuss a simple idea involving semicontinuity and how it can be applied in two different contexts to prove interesting results at the intersection of complex dynamics and geometry: to find rigid currents on K3 surfaces (work of Sibony-Verbitsky) and to study the linearization problem for neighborhoods of elliptic curves in complex surfaces with a  topologically trivial normal bundle (ongoing work with Filip).

Junyi Xie
BICMR, Peking University

Lower Semicontinuity of Dynamical Degrees

For every dominant rational self-map, we find a family of recursive inequalities of some dynamically meaningful cycles. Using these inequalities, we prove that for a family of dominant rational self-maps, the dynamical degrees are lower semi-continuous with respect to the Zariski topology.

Xinyi Yuan
Peking University

Equidistribution Over Finitely Generated Fields

The equidistribution theorem of Szpiro-Ullmo-Zhang and Yuan assumes that the base field is a number field. In Yuan-Zhang’s work on adelic line bundles, there is a conjecture on equidistribution over finitely generated fields. In this talk, Xinyi Yuan will introduce some recent progress on this conjecture.