Tangent to the Identity Germs and Affine Surfaces

10 févr. 2025, 09:00 → 14 févr. 2025, 17:00 Europe/Paris

Amphithéatre Schwartz (Institut de Mathématiques de Toulouse)

Jasmin Raissy (Institut de Mathématiques de Bordeaux), Julio Rebelo (Institut de Mathématiques de Toulouse), Lorena López Hernanz (Universidad de Alcala)

This conference will focus on the dynamics of germs tangent to the identity in several complex variables and their connections in dimension 2 with geometry of affine surfaces along with dynamical systems naturally attached to them.

There will be two basic courses in the morning:

• Charles Fougeron: affine surfaces
• Xavier Buff: local dynamics of biholomorphisms

There will be seminar talks in the evening about dynamics and about affine surfaces: 

• Matthieu Astorg
• Fabrizio Bianchi
• Corentin Boissy
• Arnaud Cheritat
• Vincent Delecroix
• Helena Reis
• Javier Ribon
• Matteo Ruggiero
• Fernando Sanz Sanchez
• Liz Vivas

 

Practical information: 

Please note that room and board for supported participants will be taken care directly by the organization. The organization will also cover the expenses for lunches (from Tuesday to Friday) and for the social dinner for all participants. Non-supported participants are expected to arrange for their own travels and accommodations in Toulouse. 

lun. 10 février

1 Affine surfaces (1)

The goal of this series of talks is to present some elements of the dynamics and topology of affine surfaces, with a focus on two subclasses: translation surfaces and dilation surfaces in the compact case.

We will begin with translation surfaces, introducing their deformation space and explaining a fundamental connection between their dynamics and the dynamics within this space, known as Masur's criterion. This criterion is a key ingredient in proving that unique ergodicity is generic for translation surfaces.

After becoming familiar with this subclass of affine surfaces, we will turn to a more general class: dilation surfaces. While these surfaces share many structural similarities with translation surfaces, their dynamics and topology generally exhibit significantly different behaviors. We will discuss their expected generic dynamics (which represent the simplest behavior from a dynamical perspective) and present a beautiful result by Veech on their triangulations.

If time permits, we will conclude by exploring some non-generic intermediate dynamical behaviors, such as the emergence of strange attractors, which can appear even in simple examples of dilation surfaces.

Orateur: Charles Fougeron

2 Local dynamics of biholomorphisms (1)

The goal of this series of lecture is to present the relation between the dynamics of germs $f:({\mathbb C}^2,0)\to ({\mathbb C}^2,0)$ tangent to the identity, the real-time dynamics of homogeneous vector fields in ${\mathbb C}^2$ and the dynamics of the geodesic flow on affine surfaces.

In the first talk, we will review the dynamics of germs $f:({\mathbb C},0)\to ({\mathbb C},0)$, in particular, the Leau-Fatou flower theorem.

Orateur: Xavier Buff (Institut de Mathématiques de Toulouse)

3 Projective structures on the n-punctured sphere and parabolic dynamics in $({\mathbb C}^3,0)$

There are at least two families of (Halphen) vector fields on ${\mathbb C}^3$ having a number of interesting properties. Among others, they induce projective structures on the 3 or 4 times punctured sphere and their dynamics is closely related to the dynamics of certain Fuchsian and Kleinian groups. Furthermore, they can be used to produce examples of tangent to the identity maps in $({\mathbb C}^3,0)$ whose dynamical study requires us to go slightly beyond the Fuchsian/Kleinian groups in question. I will try to explain this construction and say a few words about the structure of certain invariant sets.

Orateur: Helena Reis

4 Classification of polar punctures of meromorphic connections.

(Joint work with Xavier Buff) Given a meromophic connection with a pole of degree>1 near a puncture of a Riemann surface, we introduce a sequence of asymptotic values and use it to define an invariant that allows for a complete local classification of those objects, up to local conformal isomorphism. We also provide a geometric description.

Orateur: Arnaud Chéritat (CNRS/Institut de Mathéamtiques de Toulouse)

5 Horn maps of semi-parabolic Hénon maps

We prove that horn maps associated to quadratic semi-parabolic fixed points of Hénon maps, first introduced by Bedford, Smillie, and Ueda, satisfy a weak form of the Ahlfors island property. As a consequence, two natural definitions of their Julia set (the non-normality locus of the family of iterates and the closure of the set of the repelling periodic points) coincide. As another consequence, we also prove that there exist small perturbations of semi-parabolic Hénon maps for which the Hausdorff dimension of the forward Julia set J+ is arbitrarily close to 4. This is a joint work with M. Astorg.

Orateur: Fabrizio Bianchi

mar. 11 février

6 Local dynamics of biholomorphisms (2)

The goal of this series of lecture is to present the relation between the dynamics of germs $f:({\mathbb C}^2,0)\to ({\mathbb C}^2,0)$ tangent to the identity, the real-time dynamics of homogeneous vector fields in ${\mathbb C}^2$ and the dynamics of the geodesic flow on affine surfaces.

In the second lecture, we will explain how, to each germ $f:({\mathbb C}^2,0)\to ({\mathbb C}^2,0)$ tangent to the identity, one can associate a homogeneous vector field in ${\mathbb C}^2$ and a meromorphic affine surface modeled on the Riemann sphere minus finitely many points.

Orateur: Xavier Buff (Institut de Mathématiques de Toulouse)

7 Affine Surfaces (2)

The goal of this series of talks is to present some elements of the dynamics and topology of affine surfaces, with a focus on two subclasses: translation surfaces and dilation surfaces in the compact case.

We will begin with translation surfaces, introducing their deformation space and explaining a fundamental connection between their dynamics and the dynamics within this space, known as Masur's criterion. This criterion is a key ingredient in proving that unique ergodicity is generic for translation surfaces.

After becoming familiar with this subclass of affine surfaces, we will turn to a more general class: dilation surfaces. While these surfaces share many structural similarities with translation surfaces, their dynamics and topology generally exhibit significantly different behaviors. We will discuss their expected generic dynamics (which represent the simplest behavior from a dynamical perspective) and present a beautiful result by Veech on their triangulations.

If time permits, we will conclude by exploring some non-generic intermediate dynamical behaviors, such as the emergence of strange attractors, which can appear even in simple examples of dilation surfaces.

Orateur: Charles Fougeron

8 Formal transversality of the infinitesimal generator along the fixed point set

Given a tangent to the identity germ of holomorphic diffeomorphism $\phi$, we consider the map $P \mapsto \phi_{P}$ that associates to any fixed point $P$ of $\phi$ near the origin the germ $\phi_{P}$ of $\phi$ at $P$. Such germs are not in general tangent to the identity. Given the infinitesimal generator $X$ of $\phi$, a formal vector field, it is natural to ask whether we can define ``infinitesimal generators" $X_{P}$ of $\phi_{P}$ for $P \in \mathrm{Fix} (\phi)$ and if the dependence of $X_{P}$ on $P \in \mathrm{Fix} (\phi)$ is analytic. In other works, we are asking whether $X$ is formally transversal along $\mathrm{Fix} (\phi)$.

We introduce a weak and a strong concept of formal transversality for $X$. On the one hand, we will see that $X$ is always formally transversal along $\mathrm{Fix} (\phi)$ in the weak sense. On the other hand, there are examples where $X$ is not formally transversal along $\mathrm{Fix} (\phi)$ in the strong sense. We will discuss the gap between the weak and the strong concepts and provide a characterization of strong formal transversality. This is a joint work with Rudy Rosas.

Orateur: Javier Ribon

9 On translation surfaces defined by meromorphic differentials

A meromorphic one form with poles on a Riemann surface defines naturally a translation surface of infinite area. In this talk, after seeing briefly how such structures appear naturally when studying usual translation surfaces, we will describe the orbits of the geodesic flow and show how we can use this result to classify the connected components of the corresponding moduli space.

Orateur: Corentin Boissy (Université de Toulouse, IMT)

10 Local dynamics of biholomorphisms (3)

The goal of this series of lecture is to present the relation between the dynamics of germs $f:({\mathbb C}^2,0)\to ({\mathbb C}^2,0)$ tangent to the identity, the real-time dynamics of homogeneous vector fields in ${\mathbb C}^2$ and the dynamics of the geodesic flow on affine surfaces.

In the third lecture, we will study the geodesic flow of meromorphic affine surfaces modeled on compact Riemann surfaces minus finitely many points.

Orateur: Xavier Buff (Institut de Mathématiques de Toulouse)

mer. 12 février

11 Affine Surfaces (3)

The goal of this series of talks is to present some elements of the dynamics and topology of affine surfaces, with a focus on two subclasses: translation surfaces and dilation surfaces in the compact case.

We will begin with translation surfaces, introducing their deformation space and explaining a fundamental connection between their dynamics and the dynamics within this space, known as Masur's criterion. This criterion is a key ingredient in proving that unique ergodicity is generic for translation surfaces.

After becoming familiar with this subclass of affine surfaces, we will turn to a more general class: dilation surfaces. While these surfaces share many structural similarities with translation surfaces, their dynamics and topology generally exhibit significantly different behaviors. We will discuss their expected generic dynamics (which represent the simplest behavior from a dynamical perspective) and present a beautiful result by Veech on their triangulations.

If time permits, we will conclude by exploring some non-generic intermediate dynamical behaviors, such as the emergence of strange attractors, which can appear even in simple examples of dilation surfaces.

Orateur: Charles Fougeron

12 Resolution of the infinitesimal generator and parabolic manifolds in dimension 3

Through explicit examples introduced by Samuele Mongodi and myself, we will see how the resolution of singularities of vector fields of McQuillan and Panazzolo, and the resolution along separatrices of Lopez-Hernanz, Ribon, Sanz-Sanchez and Vivas, intervene in the study of parabolic manifolds for tangent to the identity germs in dimension 3.
Part of the talk is based on a work in progress with Samuele Mongodi and André Belotto Da Silva.

Orateur: Matteo Ruggiero (Université Paris Cité - IMJ-PRG)

13 Local dynamics of reduced saddle-node biholomorphisms

We study the dynamics on a full neighborhood of the origin for a biholomorphism $F$ in $\mathbb{C}^2$ that is of the reduced saddle-node type. For these type of diffeomorphisms we will show that there exist connected domains with the origin in their boundary which are either stable for $F$ or for its inverse, and that outside these domains every point is either fixed or has a finite orbit. This is a work in progress in collaboration with Lorena Lopez-Hernanz and Rudy Rosas.

Orateur: Liz Vivas

jeu. 13 février

14 Local dynamics of biholomorphisms (4)

The goal of this series of lecture is to present the relation between the dynamics of germs $f:({\mathbb C}^2,0)\to ({\mathbb C}^2,0)$ tangent to the identity, the real-time dynamics of homogeneous vector fields in ${\mathbb C}^2$ and the dynamics of the geodesic flow on affine surfaces.

In the fourth lecture, we will explain how, using the dynamics of the geodesic flow on affine surfaces, one can recover several known results on the existence of parabolic domains for germs $f:({\mathbb C}^2,0)\to ({\mathbb C}^2,0)$ tangent to the identity.

Orateur: Xavier Buff (Institut de Mathématiques de Toulouse)

15 Affine Surfaces (4)

The goal of this series of talks is to present some elements of the dynamics and topology of affine surfaces, with a focus on two subclasses: translation surfaces and dilation surfaces in the compact case.

We will begin with translation surfaces, introducing their deformation space and explaining a fundamental connection between their dynamics and the dynamics within this space, known as Masur's criterion. This criterion is a key ingredient in proving that unique ergodicity is generic for translation surfaces.

After becoming familiar with this subclass of affine surfaces, we will turn to a more general class: dilation surfaces. While these surfaces share many structural similarities with translation surfaces, their dynamics and topology generally exhibit significantly different behaviors. We will discuss their expected generic dynamics (which represent the simplest behavior from a dynamical perspective) and present a beautiful result by Veech on their triangulations.

If time permits, we will conclude by exploring some non-generic intermediate dynamical behaviors, such as the emergence of strange attractors, which can appear even in simple examples of dilation surfaces.

Orateur: Charles Fougeron

16 Parabolic manifolds of analytic diffeomorphisms along an invariant formal curve

Let $F:(\mathbb{C}^n,0)\to(\mathbb{C}^n,0)$ be a germ of a holomorphic diffeomorphism and let $\Gamma$ be a formal curve at $0$, invariant for $F$. Under certain sharp conditions on the restricted diffeomorphism $F|_\Gamma$, we show that there exists a finite non-empty family of complex submanifolds of $\mathbb{C}^n\setminus\{0\}$, invariant for $F$ and entirely composed of orbits which converge to the origin and have flat contact with $\Gamma$ (parabolic manifolds). In a second part of the talk, we adapt this result for the case of a germ of a real analytic diffeomorphism $F:(\mathbb{R}^n,0)\to(\mathbb{R}^n,0)$, where we can show, moreover, that each parabolic manifold of the family is foliated by real parabolic curves of $F$.

These results are obtained in collaboration with L. López-Hernánz, J. Ribón, J. Raissy and L. Vivas.

Orateur: Fernando Sanz

17 Singularities and flow completeness of infinite type translation surfaces

Translation surfaces are (very) particular type of affine surfaces where transition maps are translations. Though, any affine surface admits a cover which is a translation surface (possibly of infinite type). The goal of my talk is to introduce a metric invariant for singularities of infinite type translation surfaces due to Bowman-Valdez and explain how it is related to the completeness of the translation flow.

Orateur: Vincent Delecroix (CNRS, Université de Bordeaux)

18 Parabolic implosion in dimension 2

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.

Orateur: Matthieu Astorg (Université d'Orléans, IDP)

ven. 14 février

19 Affine Surfaces (5)

The goal of this series of talks is to present some elements of the dynamics and topology of affine surfaces, with a focus on two subclasses: translation surfaces and dilation surfaces in the compact case.

We will begin with translation surfaces, introducing their deformation space and explaining a fundamental connection between their dynamics and the dynamics within this space, known as Masur's criterion. This criterion is a key ingredient in proving that unique ergodicity is generic for translation surfaces.

After becoming familiar with this subclass of affine surfaces, we will turn to a more general class: dilation surfaces. While these surfaces share many structural similarities with translation surfaces, their dynamics and topology generally exhibit significantly different behaviors. We will discuss their expected generic dynamics (which represent the simplest behavior from a dynamical perspective) and present a beautiful result by Veech on their triangulations.

If time permits, we will conclude by exploring some non-generic intermediate dynamical behaviors, such as the emergence of strange attractors, which can appear even in simple examples of dilation surfaces.

Orateur: Charles Fougeron

20 Local dynamics of biholomorphisms (5)

The goal of this series of lecture is to present the relation between the dynamics of germs $f:({\mathbb C}^2,0)\to ({\mathbb C}^2,0)$ tangent to the identity, the real-time dynamics of homogeneous vector fields in ${\mathbb C}^2$ and the dynamics of the geodesic flow on affine surfaces.

In the last lecture, we will try to formulate open problems concerning the dynamics of germs $f:({\mathbb C}^2,0)\to ({\mathbb C}^2,0)$ tangent to the identity, the real-time dynamics of homogeneous vector fields in ${\mathbb C}^2$ and the dynamics of the geodesic flow on affine surfaces.

Orateur: Xavier Buff (Institut de Mathématiques de Toulouse)