Programme:
trois mini-cours sont prévus.
Juan RIVERA-LETELIER:
On the
thermodynamic formalism of rational maps (3 heures).
For a rational map f acting on the Riemann sphere and a real
parameter t , we will describe recent results on the (non-)existence of
equilibrium states of f for the
geometric potential -tln|f|, and the analytic dependence on t of
the corresponding pressure function. If time allows, we will also
discuss applications of these results to
large deviations, and to the multifractal analysis of the measure of
maximal entropy and of Lyapunov exponents.
The minicourse is naturally divided into three parts.
* The pressure function. After a quick review of the
basics of thermodynamic formalism, we consider several
characterizations of the pressure function in the specific case of
rational maps, and discuss Przytycki's version Bowen's formula in this
setting. The qualitative behavior of the pressure function
depends very much on whether the rational map satisfies the topological
Collet-Eckmann condition or not. So we will also discuss some
characterizations of this condition.
* Nice inducing schemes. We will describe the inducing
scheme associated to a "nice couple" (a kind of "backward Markov
partition"). Here we use the theory of conformal iterated function
systems developed by Mauldin and Urbanski, to obtain conditions
warranting the existence of conformal measures and equilibrium states,
as well as the analyticity of the pressure function.
* Analyticity and phase transitions. We discuss some
classes of rational maps for which the inducing scheme can be
implemented to obtain that the pressure function is analytic in the
maximal possible domain. We will show in particular how to construct
arbitrarily small nice couples for a rational map satisfying a weak
form of
hyperbolicity. In the opposite direction, we survey different known
phenomena occurring at parameters where there is no analyticity (phase
transitions), including some pathological examples where the set of
equilibrium states can be, in a precise sense, as large as possible.
Henri DE THELIN et Gabriel VIGNY:
théorème
de Yomdin
méromorphe et dynamique des applications birationnelles de CPk (4
heures).
- Première partie (De Thélin): on rappelle les
notions d'entropies topologique et métrique ainsi que la
définition des degrés dynamiques d'une application
méromorphe. Nous passons alors à la démonstration
du théorème de Yomdin et du principe variationnel. Nous
montrerons comment adapter ces idées au cadre méromorphe
pour obtenir un critère qui permet de produire des mesures
d'entropie maximale.
- Deuxième partie (Vigny): On étudie le cas
particulier d'une famille générique d'applications
birationnelles de CPk
(donnée par une condition à la
Bedford-Diller). Nous construisons pour ces applications le courant de
Green et la mesure d'équilibre. Pour pouvoir travailler en
dimension >2, on utilise la théorie des super-potentiels de
Dinh-Sibony. Nous montrons que la mesure est mélangeante et ne
charge pas les ensembles pluripolaires. En utilisant le critère
précédent, on montre que la mesure est d'entropie
maximale et qu'elle est hyperbolique.
Tomoki KAWAHIRA:
Topology of
Lyubich-Minsky's laminations for
quadratic maps: deformation and rigidity (3 heures).
Lyubich and Minsky's Riemann surface laminations and hyperbolic
3-laminations are geometric objects constructed by the dynamics of
rational maps on the Riemann
sphere. For example, we may regard the term "(quotient)
hyperbolic 3-laminations" as a possible translation of "hyperbolic
3-manifolds" for Kleinian groups in
Sullivan's dictionary. In these talks I will survey recent developments
on the topological/geometric structure of the Lyubich-Minsky
laminations of quadratic polynomials due to my collaborator C. Cabrera
and myself. Here is a list of topics:
- Construction and examples of the LM laminations.
- Deformation/Rigidity of the Riemann surface laminations
associated with hyperbolic quadratic maps.
- Riemann surface laminations associated with infinitely
renormalizable maps.
- Degeneration/Bifurcation of the LM laminations at parabolic
quadratic maps.
- An analogy: The Mandelbrot set vs the Bers slice
Pour tout renseignement ou demande d'aide financière:
Contact: favre[chez]math.jussieu.fr
Mercredi 10
Décembre |
Salle 0D1
|
12h30 - 14h00 |
Déjeuner |
14h00 - 15h00 |
DE THELIN |
15h00 - 15h30
|
Pause Café |
15h30 - 16h30
|
RIVERA |
16h30 - 17h30
|
KAWAHIRA
|
|
Jeudi 11 Décembre |
Salle 0D1 |
9h00 - 10h00
|
KAWAHIRA
|
10h00 - 10h30
|
Pause Café |
10h30 - 11h30 |
RIVERA |
11h30 - 12h30
|
DE THELIN
|
12h30 - 14h00
|
Déjeuner |
Salle 0C8 |
14h00 - 15h00
|
VIGNY
|
15h00 - 15h30
|
Pause Café |
15h30-16h30
|
RIVERA
|
|
Vendredi 12 Décembre |
Salle 0C8 |
9h30 - 10h30
|
KAWAHIRA |
10h30 - 11h00
|
Pause Café |
11h00 - 12h00 |
VIGNY
|
12h00 - 14h00
|
Déjeuner |
|
AFFICHE