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Journées de Dynamique Complexe 10, 11 et 12 Décembre 2008 Institut de Math. de Jussieu Projet ANR: Berko |
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Organisateurs: Jean-Yves Briend et Charles Favre.
Notes of Kawahira's course are online:
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).
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:
Pour tout renseignement ou demande d'aide financière:
Contact: favre[chez]math.jussieu.fr
AFFICHE
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.
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
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AFFICHE