Siegel disks, parabolic implosion and area of Julia sets
Programme
Schedule
All talks were 50mn long. The links will lead you to the presentation files if available.
Tuesday 9th
9h00 Coffee, tea, croissants
9h30 A. Douady: Introduction
10h30 J.-C. Yoccoz: Renormalisation géométrique
11h40 J. Milnor: Non locally-connected Julia sets---the infinitely renormalizable case
14h20 M. Lyubich: Hausdorff dimension and conformal measures for Feigenbaum Julia sets
15h30 C. Petersen: Deepness of the boundary of bounded type quadratic Siegel disks following C. McMullen
Wednesday 10th
9h30 L. Rempe: Control on the parabolic explosion
10h30 C. Henriksen: Area convergence from parabolics to Siegel discs
11h40 R. Oudkerk: The use of polynomial vector fields
14h20 J.H. Hubbard: Fatou coordinates and parabolic implosion
15h30 A. Epstein: Virtual Siegel disks
Thursday 11th
9h30 M. Shishikura: The parabolic/near-parabolic renormalization and an invariant class of maps
11h20 X. Buff: The boundaries of Siegel disks do not explode
14h00 A. Chéritat: Bounding the loss of measure during parabolic implosion
15h10 A. Douady: Conclusion
Thursday evening: buffet
Participants may take this opportunity to go to the "Journée COOL" at IHP, Paris, on Friday 12th (to find out where is IHP, follow this link).
Friday 12th: Journée COOL at IHP (Paris)
10h (at IHP) Yin Yongcheng: A proof of the Branner-Hubbard conjecture on Cantor Julia sets.
11h (at IHP) R. Oudkerk: The Infinite Degree Multibrot Set and Renormalization Phenomena for Meromorphic Maps with Asymptotic Values
14h (at IHP) M. Shishikura: TBA
15h45 (at IHP) L. Bartholdi: Lapins aux oreilles tordues (Travail en commun avec V. Nekrashevych.)
Short description
This celebration of Adrien's 70th birthday will focus on his successful program for Julia sets of positive area.
The problem goes back to Fatou who suggested that one should apply (to the Julia sets) the methods of Borel-Lebesgue for the measure of sets. Since then, the question remained open.
Until the 1980's, the conjecture, reinforced by the analogy with Ahlfors's conjecture on the area of limit sets of Kleinian groups, was that no Julia set of a rational function could have positive area, apart from those whose Fatou set is empty.
Results in this direction were obtained by Douady and Hubbard in the case of hyperbolic or subhyperbolic maps, by Branner, Hubbard and McMullen in the case of non renormalizable cubic polynomials with an escaping critical point, by Lyubich and Shishikura in the case of finitely renormalizable quadratic polynomials without indifferent cycles, by Petersen in the case of quadratic polynomials having a Siegel disk with bounded type rotation number.
In the 1990's, Douady began to catch a glimpse of a method for Julia sets of positive area: in the family of degree 2 polynomials with an indifferent Cremer fixed point. He put two of his Ph.D. students, Jellouli and Chéritat, on the project. By 2001, Chéritat had reduced the existence of such a Julia set to two very plausible conjectures.
Using work of McMullen on Siegel disks with bounded rotation numbers and work of Inou and Shishikura on parabolic renormalization, Buff (another descendent of Douady) and Chéritat proved recently those conjectures and thus brought Douady's method to completion.
This conference will concentrate on the many tools used in the proof. Among them, Parabolic Implosion and Perturbation of Siegel disks are of main importance. Nonetheless important are Quasiconformal Models, Sector Renormalization, Cylinder Renormalization, Control on Parabolic Explosion and the Deepness of constant type Siegel disks. We invited world's leading experts in these questions.
The problem goes back to Fatou who suggested that one should apply (to the Julia sets) the methods of Borel-Lebesgue for the measure of sets. Since then, the question remained open.
Until the 1980's, the conjecture, reinforced by the analogy with Ahlfors's conjecture on the area of limit sets of Kleinian groups, was that no Julia set of a rational function could have positive area, apart from those whose Fatou set is empty.
Results in this direction were obtained by Douady and Hubbard in the case of hyperbolic or subhyperbolic maps, by Branner, Hubbard and McMullen in the case of non renormalizable cubic polynomials with an escaping critical point, by Lyubich and Shishikura in the case of finitely renormalizable quadratic polynomials without indifferent cycles, by Petersen in the case of quadratic polynomials having a Siegel disk with bounded type rotation number.
In the 1990's, Douady began to catch a glimpse of a method for Julia sets of positive area: in the family of degree 2 polynomials with an indifferent Cremer fixed point. He put two of his Ph.D. students, Jellouli and Chéritat, on the project. By 2001, Chéritat had reduced the existence of such a Julia set to two very plausible conjectures.
Using work of McMullen on Siegel disks with bounded rotation numbers and work of Inou and Shishikura on parabolic renormalization, Buff (another descendent of Douady) and Chéritat proved recently those conjectures and thus brought Douady's method to completion.
This conference will concentrate on the many tools used in the proof. Among them, Parabolic Implosion and Perturbation of Siegel disks are of main importance. Nonetheless important are Quasiconformal Models, Sector Renormalization, Cylinder Renormalization, Control on Parabolic Explosion and the Deepness of constant type Siegel disks. We invited world's leading experts in these questions.