Schedule
| Monday May 2nd |
| 7:15 - 9:00 |
Breakfast |
| 9:00 - 9:45 |
|
| 10:00 - 10:45 |
|
| 10:45 - 11:00 |
Coffee break |
| 11:00 - 11:45 |
|
| 12:15 - 13:15 |
Lunch |
| 13:15 - 15:45 |
Free time |
| 15:45 - 16:00 |
Coffee break |
| 17:00 - 17:45 |
|
| 18:00 - 18:45 |
|
| 19:15 - 20:15 |
Dinner |
| Tuesday 3rd |
| 7:15 - 9:00 |
Breakfast |
| 9:00 - 9:45 |
|
| 10:00 - 10:45 |
|
| 10:45 - 11:00 |
Coffee break |
| 11:00 - 11:45 |
|
| 12:15 - 13:15 |
Lunch |
| 13:15 - 15:45 |
Free time |
| 15:45 - 16:00 |
Coffee break |
| 17:00 - 17:45 |
|
| 18:00 - 18:45 |
|
| 19:15 - 20:15 |
Dinner |
| Wednesday 4th |
| 7:15 - 9:00 |
Breakfast |
| 10:00 - 10:45 |
|
| 11:00 - 11:45 |
|
| 10:45 - 11:00 |
Coffee break |
| 12:15 - 13:15 |
Lunch |
| 13:15 - 19:00 |
Free afternoon |
| 19:15 - 20:15 |
Dinner |
| Thursday 5th |
| 7:15 - 9:00 |
Breakfast |
| 9:00 - 9:45 |
|
| 10:00 - 10:45 |
|
| 10:45 - 11:00 |
Coffee break |
| 11:00 - 11:45 |
|
| 12:15 - 13:15 |
Lunch |
| 13:15 - 15:45 |
Free time |
| 15:45 - 16:00 |
Coffee break |
| 17:00 - 17:45 |
|
| 18:00 - 18:45 |
|
| 19:15 - 20:15 |
Dinner |
| Friday 6th |
| 7:15 - 9:00 |
Breakfast |
| 9:00 - 9:45 |
|
| 10:00 - 10:45 |
|
| 10:45 - 11:00 |
Coffee break |
| 11:00 - 11:45 |
|
| 12:15 - 13:15 |
Lunch |
| 13:15 - 15:45 |
Free time |
| 15:45 - 16:00 |
Coffee break |
| 17:00 - 17:45 |
|
| 18:00 - 18:45 |
|
| 19:15 - 20:15 |
Dinner |
Abstracts
Marie-Claude Arnaud
Regularity of the invariant curves of conservative twist maps
Abstract of part 1
The conservative twist maps were introduced by H.Poincaré at the end of the 19th century for the study of the circular restricted 3 Body problem. Then Birkhoff began the study of these maps and obtained various theorems concerning their invariant curves. Fifty years later, the famous K.A.M. theorems provided new (perturbative) results for these invariant curves. We will give a survey concerning old and new results, and open probems.
Abstract of part 2
Answering to a question of J. Mather, we will construct an example of a C1 conservative twist map that has an essential irrational invariant curve that is not differentiable.
Xavier Buff
Linearisability in holomorphic dynamics.
I will present a proof by Yoccoz of the following result originally due to Siegel: there is a set S of full measure in the unit circle such that for all ρ in S, any holomorphic germ z → ρ z + O(z2) is conjugate to the rotation z → ρ z near the origin. The proof relies on a clever use of harmonic and subharmonic functions, rather than on the arithmetic properties of the rotation number (as in the original proof of Siegel). I will then present further developments due to Ricardo Perez-Marco and other developments due to Arnaud Chéritat and myself.
Arnaud Chéritat
Siegel disks
The first part is a survey on Siegel disks. Siegel disks are maximal domains on which a given (one complex dimensional) dynamical system is conjugated to a rotation.
In the second part I will prove a particular theorem: there exist injective holomorphic maps, defined on arbitrarily large disks, having Siegel disks contained in the unit disk, and whose boundary is not locally connected. The construction is an adaptation of the Anosov-Katok method.
Christophe Dupont
Dimension theory in holomorphic dynamics
We deal with the dynamics of holomorphic mappings acting on projective spaces Pk(C).
Our aim is to study the size (Hausdorff dimension) δ of their maximal entropy measure μ.
First talk :
1) We begin by focusing on k=1: we study rational maps acting on the Riemann sphere.
- We present Mané's formula, which expresses δ in terms of the metric entropy and the Lyapunov exponent of μ. The proof heavily relies on the conformal property of rational maps.
- We also focus on the maximal case δ = 2: those mappings are characterized by a strong algebraic rigidity.
2) We consider k=2: here dimension properties are less understood since the mappings are no more conformal.
- We present Binder-de Marco's conjecture concerning the dimension δ of μ. We then state known estimates in view of that formula.
- The maximal case δ = 4 has been also entirely characterized. We shall see that the higher dimensional context k=2 requires new tools.
Second talk :
We sketch the proof of :
1) k=1 : Mané's formula.
2) k=2 : the lower estimate in Binder-de Marco's conjecture.
Charles Favre
One dimensional non archimedean dynamics
The main goal is to describe some of the basic results in the study of the
iteration of rational functions defined over a non archimedean field. This
theory is quite new and presents many similarities with the complex case,
although being much more algebraic in nature. We shall also try to explain some
of the motivations that lead to such studies.
Jacques Fejoz
Moser's normal form theorem of vector fields in the neighborhood
of a Diophantine invariant torus, and consequences in KAM theory
The method of parameters reduces the persistence problem of many kinds
of objects, to finite dimensional problems. This is true in particular
for Diophantine invariant tori, in the theory of dynamical
systems. This method can be traced back to Poincaré's proof of the
existence of Linstedt series in Hamiltonian systems, at the formal
level, and to a remarquable normal form theorem of Moser at the
analytic level. Many invariant tori theorems of Kolmogorov, Arnold,
Rüssmann, Herman and others can be deduced from Moser's normal
form. We will present Moser's theorem and some of its consequences. In
the second lecture we will sketch some ideas of the proofs.
Gabriel Paternain
Transparent connections
I will discuss the inverse problem of reconstructing a connection from knowledge about its parallel transport.
A connection is said to be transparent if its parallel
transport along every closed geodesic is the identity.
I will explain how to describe all transparent
SU(2)-connections over a closed negatively curved surface.
This needs the Livsic theorem for cocycles over the geodesic flow with values in a compact Lie group and a suitable Bäcklund transformation that exploits the holomorphic structure induced
by the connection.
If time permits I will also discuss transparent pairs (A,Φ),
where A is a connection and Φ is a Higgs field.
Ricardo Pérez Marco
Applications of potential theory to holomorphic and hamiltonian dynamics
We will recall some basic potential and pluripotential
theory, and the application to the total convergence or
generic divergence of problems of small divisors in holomorphic
families of holomorphic dynamical systems. The same tools will be
applied to the problem of integrability of a hamiltonian system in the
neighborhood of an equilibrium point, and to the convergence of
Birkhoff normal form.