Journées de Dynamique Complexe
10, 11 et 12 D
écembre 2008
Institut de Math. de Jussieu
Projet ANR: Berko

                      Organisateurs: Jean-Yves Briend et Charles Favre.

Notes of Kawahira's course are online:
  1. First talk
  2. Second talk
  3. Third talk

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).
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:
  1. Construction and examples of the LM laminations.
  2. Deformation/Rigidity of the Riemann surface laminations associated with hyperbolic quadratic maps.
  3. Riemann surface laminations associated with infinitely renormalizable maps.
  4. Degeneration/Bifurcation of the LM laminations at parabolic quadratic maps.
  5. An analogy: The Mandelbrot set vs the Bers slice

Pour tout renseignement ou demande d'aide financière:
: favre[chez]

Mercredi 10 Décembre
Salle 0D1
12h30 - 14h00 Déjeuner
14h00 - 15h00 DE THELIN
15h00 - 15h30
Pause Café
15h30 - 16h30
16h30 - 17h30

Jeudi 11 Décembre
Salle 0D1
 9h00 - 10h00
10h00 - 10h30
Pause Café
10h30 - 11h30 RIVERA
11h30 - 12h30
12h30 - 14h00
Salle 0C8
14h00 - 15h00
15h00 - 15h30
Pause Café

Vendredi 12 Décembre
Salle 0C8
 9h30 - 10h30
10h30 - 11h00
Pause Café
11h00 - 12h00 VIGNY
12h00 - 14h00