On the dynamics of tangent-like mappings

• Nuria Fagella

Room: Amphithéatre Schwartz 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 maps with a free asymptotic value. Likewise, in parameter space, we find copies of the "Mandelshell", the universal object whose boundary is the bifurcation locus of the family $T_\alpha$. The concept of "tangent-like mappings" was originally defined by Galazka and Kotus in 2008. This is joint work (in progress) with Anna Miriam Benini and Matthieu Astorg.

Parabolic implosion in dimension 2

• Matthieu Astorg (Université d'Orléans, IDP)

Room: Amphithéatre Schwartz 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 work with Lorena Lopez-Hernanz and J. Raissy.

A priori bounds for some near-parabolic primitive combinatorics

• Alex Kapiamba

Room: Amphithéatre Schwartz 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.

Boundaries of multiply connected Fatou components. A unified approach

• Anna Jové

Room: Amphithéatre Schwartz 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 can deal with all Fatou components (invariant or wandering, with all possible internal dynamics) simultaneously. We will explore the similarities and the differences that appear for Fatou components of transcendental functions (both invariant and wandering) in contrast with rational maps. This is joint work with G. R. Ferreira.

Towards Transcendental Thurston Theory

• Nikolai Prochorov (Université d'Aix-Marseille)

Room: Amphithéatre Schwartz 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 equivalent (in a certain dynamical sense) to a holomorphic one. The result of W. Thurston and further developments allow us to answer this question quite effectively in the setting of (postcritically finite) maps of finite degree, and it has numerous applications for the dynamics of rational maps. A similar question can be formulated for the maps of infinite degrees (i.e., in the transcendental setting), for instance, for entire or meromorphic postsingularly finite maps. However, the characterization problem becomes significantly more complicated, and the complete answer in the transcendental case is still not known. The first breakthrough in this area was achieved by J.H. Hubbard, D. Schleicher, and M. Shishikura, who provided a topological characterization of postsingularly finite exponential maps. Although this family is relatively simple, their result required the development of entirely new techniques. In my talk, I am going to introduce key notions of Thurston's theory in the transcendental setting. I will present a result demonstrating that a variant of Thurston's theorem applies to a broad class of transcendental maps, many of which are not defined by simple explicit formulas. If time allows, I will also briefly discuss a "relative" version of Thurston's theorem, which holds in full generality for both finite and infinite degree cases.

Dynamics of toric rational maps

• Roland Roeder

Room: Amphithéatre Schwartz 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 algebraically stable models. In this talk I will explain my joint work with Jeffrey Diller investigating the equidistribution and ergodic properties of these mysterious mappings.

Compactifications of moduli spaces and strata of differentials

• Benjamin Dozier

Room: Amphithéatre Schwartz 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 several later talks at this conference. We will take a tour of these compactifications, discussing constructions, examples, and applications, mainly from the perspective of flat geometry. In particular, we will consider the "What You See Is What You Get" compactification studied by Mirzakhani-Wright, and the Multi-scale compactification constructed by Bainbridge-Chen-Gendron-Grushevsky-Moller.

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

• Myeongjae Lee

Room: Amphithéatre Schwartz 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.

Algebraic degrees of stretch factors of pseudo-Anosov maps

• Erwan Lanneau (Institut Fourier)

Room: Amphithéatre Schwartz 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 challenge in the theory. Thurston, in his paper explaining his famous construction of products of multitwists (popularized by a talk of John Hubbard after a bouillabaisse at CIRM) claimed, without proof, that the pseudo-Anosov maps obtained by this construction show that the bound 6g-6 is sharp. I will explain how to justify this claim and show that every even degree between 2 and 6g-6 arises as the stretch factor degree of a pseudo-Anosov mapping class in the Torelli group.

Approximating Transcendental Thurston Theory

• Malavika Mukundan

Room: Amphithéatre Schwartz 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.

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

• Eric Bedford

Room: Amphithéatre Schwartz 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.

Free time for discussions Room: Amphithéatre Schwartz

Rescaling Limits of Rational Maps

• Jan Kiwi

Room: Amphithéatre Schwartz 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.

The locus of matings and captures in Per_n(0)

• Caroline Davis

Room: Amphithéatre Schwartz Matings and captures are ways to relate decompose a priori complicated ``rational maps” into well-understood``polynomial” 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.

Teapots and entropy algorithms for the Mandelbrot set

• Kathryn Lindsey

Room: Amphithéatre Schwartz 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 eigenvalues outside the unit circle change continuously with external angle, while those within the unit circle exhibit "persistence" along principal veins. To establish these results, we developed a version of kneading theory adapted to principal veins and proved the equivalence of multiple core entropy algorithms. In this talk, I will discuss this circle of ideas, emphasizing how entropy provides a lens on the geometry of the Mandelbrot set.

Volumes of odd strata of quadratic differentials

• Elise Goujard (IMB)

Room: Amphithéatre Schwartz 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 count of integer metrics on ribbon graphs with prescribed odd valencies. The study of the possible degenerations of these ribbon graphs allows to express the difference between the volume of the "completed" stratum and the volume of the stratum as a linear combination of volumes of boundary strata, with explicit rational coefficients. Several conjectures on the large genus asymptotics of volumes or distribution of cylinders follow from this formula. (joint work with E. Duryev and I. Yakovlev).

Counting in polygonal billiards

• Pascal Hubert

Room: Amphithéatre Schwartz 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 trajectories and count the number of codes of a given length. I will explain the relation between these two problems and give very recent results obtained with Jayadev Athreya and Serge Troubetzkoy.

Parabolic implosion in the parameter space of cubic polynomials

• Runze Zhang

Room: Amphithéatre Schwartz 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 bifurcation loci occurs. We investigate the topology of such enrichment in the parameter space and relate it to the corresponding enrichment of Julia sets of quadratic polynomials, the latter of which has been studied systematically by P. Lavaurs in the 80s.

Connected components of strata of abelian differentials

• Martin Möller

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

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

• Vincent Delecroix (CNRS - Université de Bordeaux)

Room: Amphithéatre Schwartz 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 the talk are the $\operatorname{GL}^+_2(\mathbb{R})$-action on the moduli space of translation surfaces and the multi-scale compactification of $\operatorname{GL}^+_2(\mathbb{R})$-orbit closures. After presenting the relevance of $\operatorname{GL}^+_2(\mathbb{R})$-orbit closures in the understanding of linear flows, I will describe how these objects are amenable to efficient computations (in the sense of computer programs). This talk will be based on joint works with Julian Rüth, Kai Fu and Bradley Zykoski.

Topology of Henon maps

• Tanya Firsova

Room: Amphithéatre Schwartz

On the Hubbard program for Hénon mappings

• Raluca Tanase

Room: Amphithéatre Schwartz 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 ${\mathbb C}^2$.

Transcendental Thurston Theory

• Dierk Schleicher

Room: Amphithéatre Schwartz 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.

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

• Mitsuhiro Shishikura

Room: Amphithéatre Schwartz 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 Berkovich space of the extension of the field of Laurent series. This can be considered as a ``spider algorithm’’ in the Berkovich space. This is a unfinished work in progress with Arfeux and Kiwi, and some work-out examples with E. Hironaka related to Per_n(0) and Rohini Ramadas.