JHH80 Dynamical Developments: Degenerations of Flat Surfaces and Rational Maps

Contribution List

1. On the dynamics of tangent-like mappings

Nuria Fagella

6/10/25, 9:00 AM

In this talk we will introduce a transcendental version of the theory of polynomial-like mappings. The model family is a one parameter family $T_\alpha$ of "generalized tangent maps", which are meromorphic funtions with exactly two asymptotic values, only one of which is free. A straightenning theorem will explain why we find copies of Julia sets of $T_\alpha$ in the dynamical plane of other...

2. Parabolic implosion in dimension 2

Matthieu Astorg (Université d'Orléans, IDP)

6/10/25, 10:20 AM

Parabolic implosion is a tool for studying the dynamics of perturbations of a map with a fixed point tangent to the identity, or more generally with at least one eigenvalue which is a root of unity. We will start by surveying classical parabolic implosion in dimension one, and then we will explain an ongoing work on parabolic implosion of germs tangent to the identity in dimension 2.

Joint...

23. A priori bounds for some near-parabolic primitive combinatorics

Alex Kapiamba

6/10/25, 11:30 AM

The local connectivity of the Mandelbrot set (MLC) is a long outstanding conjecture in complex dynamics. Nearly twenty years ago, Kahn and Lyubich established MLC for all “definitely primitive” combinatorics. In this talk we will discuss MLC for some primitive combinatorics which accumulate on parabolic parameters in the Mandelbrot set. Based on joint work with Jeremy Kahn.

3. Boundaries of multiply connected Fatou components. A unified approach

Anna Jové

6/10/25, 2:00 PM

In this talk, we will focus on boundaries of multiply connected Fatou components, from a topological, measure-theoretical and dynamical point of view. The main tool in our analysis is the universal covering map (and its boundary extension), which allows us to relate the dynamics on the boundary with the dynamics of the radial extension of the so-called associated inner function. This way, we...

4. Towards Transcendental Thurston Theory

Nikolai Prochorov (Université d'Aix-Marseille)

6/10/25, 3:00 PM

In the 1980’s, William Thurston obtained his celebrated characterization of post-critically finite rational maps. This result laid the foundation of such a field as Thurston's theory in holomorphic dynamics, which has been actively developing in the last few decades. One of the most important problems in this area is the characterization question, which asks whether a given topological map is...

5. Dynamics of toric rational maps

Roland Roeder

6/10/25, 4:15 PM

I will describe a class of rational maps in two complex variables that preserve the meromorphic two form $\eta = dx \wedge dy / (xy)$. This property makes their dynamics easier to study, while still providing rich examples. Indeed, the mappings that were recently proved by Bell-Diller-Jonsson to have transcendental dynamical degree preserve $\eta$. Such mappings do not admit...

6. Compactifications of moduli spaces and strata of differentials

Benjamin Dozier

6/11/25, 9:00 AM

In this expository survey talk, we will begin by recalling the Deligne-Mumford compactification of the moduli space of Riemann surfaces of fixed genus. Building on this, we will discuss the topic of compactifications of strata of differentials on Riemann surfaces (i.e. spaces of translation surfaces), which has seen intense activity in the last decade, and which will play an important role in...

7. Connected components of the generalized strata of meromorphic differentials with linear residue conditions

Myeongjae Lee

6/11/25, 10:25 AM

Generalized strata are linear submanifolds of strata of meromorphic differentials, defined as subloci where certain sets of residues of the poles sum up to zero. We classify the connected components of the generalized strata, with a degeneration technique to the boundary of the multi-scale compactification. This is a joint work with Yiu Man Wong.

8. Algebraic degrees of stretch factors of pseudo-Anosov maps

Erwan Lanneau (Institut Fourier)

6/11/25, 11:25 AM

An important aspect of the theory of pseudo-Anosov mapping
classes concerns the study of the stretch factor lambda(f) of a
pseudo-Anosov mapping class f. This is a bi-Perron algebraic integer
of degree bounded above by 6g-6 which is the dimension of the
Teichmüller space for the underlying surface. The question of
realising any bi-Perron algebraic integer as a stretch factor is a
major...

9. Approximating Transcendental Thurston Theory

Malavika Mukundan

6/11/25, 2:00 PM

We will discuss emerging trends in the study of transcendental Thurston maps, beginning with known results on realization criteria. Our work in progress attempts to realize several objects in transcendental Thurston theory as 'limits' of corresponding objects from the Thurston polynomial setting. We explore the connection between this program and a Thurston-type classification and related problems.

10. Dynamics of rational maps in dimension 2 and the renormalization map for the iterated monodromy group of the Basilica map

Eric Bedford

6/11/25, 3:00 PM

We will look at the dynamics of a family of birational maps on R^2 and C^2. Then we will discuss a noninvertible rational map acting in dimension 2. This map arises as the renormalization map in the Iterated Monodromy group of the Basilica map.

24. Free time for discussions

6/11/25, 4:00 PM

11. Rescaling Limits of Rational Maps

Jan Kiwi

6/12/25, 9:00 AM

As rational maps degenerate, the simplest limiting dynamical systems that arise are known as "rescaling limits". In this survey talk, we will discuss rescaling limits and related ideas, as well as some applications to degenerate rational dynamics.

12. The locus of matings and captures in Per_n(0)

Caroline Davis

6/12/25, 10:25 AM

Matings and captures are ways to relate decompose a priori complicated rational maps” into well-understoodpolynomial” dynamics. We present applications of various models for the locus of matings and captures in Per_n(0) towards problems like irreducibility of Per_n(0) and locating non-matings and punctures.

13. Teapots and entropy algorithms for the Mandelbrot set

Kathryn Lindsey

6/12/25, 11:25 AM

Thurston’s "Master Teapot" is a three-dimensional fractal-like object that captures how topological entropy varies for real quadratic polynomials. In joint work with Chenxi Wu and Giulio Tiozzo, we constructed an analogous "teapot" for each principal vein of the Mandelbrot set, extending key geometric properties from the real setting to the complex plane. Specifically, we showed that...

14. Volumes of odd strata of quadratic differentials

Elise Goujard (IMB)

6/12/25, 2:00 PM

I will present a formula giving the Masur-Veech volumes of ”completed” odd strata of quadratic differentials as a sum over stable graphs. This formula generalizes Delecroix-G-Zograf-Zorich formula in the case of principal strata. The coefficients of the formula are in this case intersection numbers of psi classes with the Witten-Kontsevich combinatorial classes; they naturally appear in the...

15. Counting in polygonal billiards

Pascal Hubert

6/12/25, 3:00 PM

Counting the periodic trajectories of length at most T in a polygonal billiard goes back to Gauss (in the square, it is the famous Gauss circle problem). If the angles of the polygon are rational, several important results by Masur, Veech, Eskin-Masur, Eskin-Mirzakhani-Mohammadi give estimates on the number of periodic orbits when the length tends to infinity. One can also code the billiard...

16. Parabolic implosion in the parameter space of cubic polynomials

Runze Zhang

6/12/25, 4:15 PM

Parabolic implosion is a remarkable phenomenon in complex dynamics. It describes the enrichment of Julia sets when the parabolic point of a rational map is perturbed. It is also natural to study the parabolic implosion in parameter spaces. In particular, when one perturbs properly the family of cubic polynomials having a stable parabolic fixed point into nearby families, the enrichment of...

17. Connected components of strata of abelian differentials

Martin Möller

6/13/25, 9:00 AM

We revisit this classical topic in the geometry of strata with an eye on arbitrary characteristic, using recent advances for compactifying strata

18. Computation of linear subvarieties in the moduli space of translation surfaces and their multi-scale boundary

Vincent Delecroix (CNRS - Université de Bordeaux)

6/13/25, 10:25 AM

A translation surface is a surface endowed with
an atlas whose change of charts are translations. Fundamental
examples include the flat tori $\mathbb{C} / \Lambda$. A translation
surface comes with a one-parameter family of linear flows, one
for each direction in $\mathbb{C}$. Translation surfaces naturally
appear when considering billiard flows in rational polygons.

The main focus of...

19. Topology of Henon maps

Tanya Firsova

6/13/25, 11:25 AM

20. On the Hubbard program for Hénon mappings

Raluca Tanase

6/13/25, 2:00 PM

In this talk we will present some results and research directions of John Hubbard which have been influential for the study of the dynamics of polynomial automorphisms of ${\mathbb C}^2$.

21. Transcendental Thurston Theory

Dierk Schleicher

6/13/25, 3:00 PM

We discuss how to extend Thurston’s famous characterization theorem of rational maps to a natural and interesting class of transcendental maps. This is joint work work with Sergey Shemyakov and based on his PhD thesis, as well as extensions thereof.

22. Rescaling limit of quadratic rational maps and a search for Berkovich spider

Mitsuhiro Shishikura

6/13/25, 4:15 PM

When a family of rational maps degenerates, certain parametrized coordinate changes may give rise to a non-trivial return map. J. Kiwi studied such scaling limits for quadratic rational maps and M. Arfeux defined ``trees of spheres’’ for the degeneration. We will discuss a converse problem which means a construction of degeneration family from a given data, and its relation to the...