Tangent to the Identity Germs and Affine Surfaces

Liste des Contributions

1. Affine surfaces (1)

Charles Fougeron

10/02/2025 09:00

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...

2. Local dynamics of biholomorphisms (1)

Xavier Buff (Institut de Mathématiques de Toulouse)

10/02/2025 10:30

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...

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

Helena Reis

10/02/2025 14:00

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...

4. Classification of polar punctures of meromorphic connections.

Arnaud Chéritat (CNRS/Institut de Mathéamtiques de Toulouse)

10/02/2025 15:00

(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.

18. Horn maps of semi-parabolic Hénon maps

Fabrizio Bianchi

10/02/2025 16:15

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...

6. Local dynamics of biholomorphisms (2)

Xavier Buff (Institut de Mathématiques de Toulouse)

11/02/2025 09:00

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 $f:({\mathbb C}^2,0)\to ({\mathbb C}^2,0)$...

7. Affine Surfaces (2)

Charles Fougeron

11/02/2025 10:30

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...

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

Javier Ribon

11/02/2025 14:00

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...

9. On translation surfaces defined by meromorphic differentials

Corentin Boissy (Université de Toulouse, IMT)

11/02/2025 15:00

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.

10. Local dynamics of biholomorphisms (3)

Xavier Buff (Institut de Mathématiques de Toulouse)

11/02/2025 16:05

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...

11. Affine Surfaces (3)

Charles Fougeron

12/02/2025 09:00

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...

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

Matteo Ruggiero (Université Paris Cité - IMJ-PRG)

12/02/2025 10:20

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...

13. Local dynamics of reduced saddle-node biholomorphisms

Liz Vivas

12/02/2025 11:20

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....

14. Local dynamics of biholomorphisms (4)

Xavier Buff (Institut de Mathématiques de Toulouse)

13/02/2025 09:00

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,...

15. Affine Surfaces (4)

Charles Fougeron

13/02/2025 10:30

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...

16. Parabolic manifolds of analytic diffeomorphisms along an invariant formal curve

Fernando Sanz

13/02/2025 14:00

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...

17. Singularities and flow completeness of infinite type translation surfaces

Vincent Delecroix (CNRS, Université de Bordeaux)

13/02/2025 15:00

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...

5. Parabolic implosion in dimension 2

Matthieu Astorg (Université d'Orléans, IDP)

13/02/2025 16:15

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...

19. Affine Surfaces (5)

Charles Fougeron

14/02/2025 09:00

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...

20. Local dynamics of biholomorphisms (5)

Xavier Buff (Institut de Mathématiques de Toulouse)

14/02/2025 10:30

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...