Programme
| Lundi 26 mai | |
| 9:00 - 9:30 | Accueil des participants |
| 9:30 - 10:45 | |
| 10:45 - 11:00 | Pause café |
| 11:00 - 12:00 | Mikhail Lyubich
Dynamics of unicritical polynomials |
| 12:00 - 14:00 | Déjeuner d'accueil à l'IHP |
| 14:00 - 15:00 | |
| 15:00 - 15:30 | Pause café |
| 15:30 - 16:30 | John Milnor
Critically Periodic Cubic Polynomials |
| 16:30 - 16:45 | Pause |
| 16:45 - 17:45 | Peter Haissinsky
Random walks on hyperbolic groups |
| 18:00 - 18:30 | François Tisseyre/EcoutezVoir |
| Mardi | |
| 9:30 - 10:30 | Adam L. Epstein
La thèse de Douady |
| 10:30 - 11:00 | Pause café |
| 11:00 - 12:00 | Jean-Pierre Demailly
Intrinsic metrics and solutions of Monge-Ampère equations |
| 14:00 - 15:00 | |
| 15:00 - 15:30 | Pause café |
| 15:30 - 16:30 | |
| 16:30 - 17:00 | Pause |
| 17:00 - 20:00 | Souvenirs d'Adrien |
| Mercredi | |
| 9:30 - 10:30 | |
| 10:30 - 11:00 | Pause café |
| 11:00 - 12:00 | Dennis Sullivan
String configurations in dimensions 2,3,4,... |
| 14:00 - 15:00 | |
| 15:00 - 15:30 | Pause café |
| 15:30 - 16:30 | |
| 16:30 - 16:45 | Pause |
| 16:45 - 17:45 | Vladimir Arnold
Random and algebraic permutations' statistics |
| 18:00 - 19:30 | Posters |
| Jeudi | |
| 9:30 - 10:30 | Mikhail Gromov
Entropy and Linearized Isoperimetry |
| 10:30 - 11:00 | Pause café |
| 11:00 - 12:00 | |
| 14:00 - 15:00 | Sebastian van Strien
On the interface of real and holomorphic dynamics |
| 15:00 - 15:30 | Pause café |
| 15:30 - 16:30 | |
| 16:30 - 17:00 | Pause |
| Soirée | Buffet + concert à l'École Normale Supérieure |
| Vendredi | |
| 9:30 - 10:30 | Genadi Levin
Multipliers of periodic orbits for holomorphic maps |
| 10:30 - 11:00 | Pause café |
| 11:00 - 12:00 | |
| 14:00 - 15:00 | |
| 15:00 - 15:30 | Pause café |
| 15:30 - 16:30 | |
| 16:30 - 16:45 | Pause |
| 16:45 - 17:45 | |
Résumés
Vladimir Arnold Random and algebraic permutations' statistics
The period T(A) of a permutation A of N points might attain large values even when N is not so large: there exist permutations A∈S(100) of N=100 points whose periods are greater than 230 000 000.
A matrix A in SL(2,ℤ) defines a permutation  in S(n2) of the N=n2 points of the finite torus ℤn2, and a permutation Ã∈S(nK(n)) of the N=nK(n) points of the finite projective line ℙ1(ℤn), where K(n)=∏(1+1/p), (p being the prime divisors of n).
The periods of these algebraic permutations Â, Ã are much smaller than for the generic permutations of the same points.
The American physicist F. Dyson has discovered that for the "Arnold's cat mapping"
the period of  in S(22500) is only T(Â)=300, studying experimentally the computerized version of the Arnold's cat mapping on a display of n=150 pixels, ℤn2.
The statistics of the Young diagrams of cycles of the algebraic permutations  and à is very different from that of the generic permutations of the same number N of points (whose study started from the 1944 paper by V.L. Goutchazov).
For instance the mean number of cycles of a random permutation grows with the number N of the permuted points like log N. The number of the length s cycles of all the N! permutations forming S(N), equals N!/s. It means that the belonging of a point to the cycles of any length 1≤s≤N have equal probabilities.
Among other results of these studies of the statistics of the random and of the algebraic permutations is the following strange Theorem.
The Cesaro mean values K̂ of the numbers K(n) tend, as n tends to ∞, to a finite limit K̂(∞)=lim 1/n ∑m=1n K(m) = 15/π2. This theorem, deduced from the empirical observation of the coincidence of 20 first digits, is now proved, using the formula K̂(∞)=ζ(2)/ζ(4)
The periods of these algebraic permutations Â, Ã are much smaller than for the generic permutations of the same points.
The American physicist F. Dyson has discovered that for the "Arnold's cat mapping"
| A = |
|
The statistics of the Young diagrams of cycles of the algebraic permutations  and à is very different from that of the generic permutations of the same number N of points (whose study started from the 1944 paper by V.L. Goutchazov).
For instance the mean number of cycles of a random permutation grows with the number N of the permuted points like log N. The number of the length s cycles of all the N! permutations forming S(N), equals N!/s. It means that the belonging of a point to the cycles of any length 1≤s≤N have equal probabilities.
Among other results of these studies of the statistics of the random and of the algebraic permutations is the following strange Theorem.
The Cesaro mean values K̂ of the numbers K(n) tend, as n tends to ∞, to a finite limit K̂(∞)=lim 1/n ∑m=1n K(m) = 15/π2. This theorem, deduced from the empirical observation of the coincidence of 20 first digits, is now proved, using the formula K̂(∞)=ζ(2)/ζ(4)
Bodil Branner Polynomial vector fields in one complex variable
In recent years Adrien Douady was interested in polynomial vector fields, both in relation to iteration theory and as a topic on their own. This talk is based on his work with Pierrette Sentenac, work of Xavier Buff and Tan Lei, and my own joint work with Kealey Dias.
César Camacho The moduli of C*-actions on Stein 2-dimensional analytic spaces
We will show the analytic classification of C*-actions, with at least one
fixed point, defined on a Stein 2-dimensional variety. In the dicritical
situation the moduli of these actions can be
established in terms of the resolution of the singularity. In the
nondicritical case a generic
action is the pull-back, by means of a meromorphic mapping, of a simple
model.
Arnaud Chéritat The quest for positive measure Julia sets
We will explain our joint work with Xavier Buff on
Douady's programme that led to the proof of existence of
positive measure Julia sets.
Jean-Pierre Demailly Intrinsic metrics and solutions of Monge-Ampère equations
Following a number of suggestions made by Y.Kawamata and more recently by
H.Tsuji, we will present a construction of intrinsic Kähler currents
on non uniruled projective varieties and study some of their basic
properties. The link with the solution of certain degenerate
Monge-Ampère equations will also be discussed.
Adam L. Epstein La thèse de Douady
Soit X un espace analytique complexe. Let but de cette thèse était de munir son auteur du grade de docteur es-sciences mathématiques, et l'ensemble H(X) des sous-espaces analytiques compacts de X d'une structure d'espace analytique. Nous discuterons du deuxième problème.
Nuria Fagella Surgery and the limbs of the Mandelbrot set
After the first occurrences of surgery at the beginning of the eighties,
Douady and Branner in 1987 went one step further and used a surgery
construction to relate polynomials of a different degree defining
in this fashion a homeomorphism between (subsets of) connectedness
loci of different families of polynomials. They had to deal with delicate
issues like the continuity of surgery procedures with respect to
parameters. In this talk we will sketch their construction and techniques
and discuss how these have been used later to obtain other interesting
results.
Charles Favre Birational classification of rational dynamical systems of ℙ2
Studying dynamical systems induced by rational maps of ℙ2
naturally leads to the question of construction of invariants
modulo birational conjugacy. The most basic one is given
by the growth of degrees of the iterates of the map. One shall review
the known results on the sequence of degrees focusing on the polynomial
case. This is a joint work with S. Boucksom and M. Jonsson.
Peter Haissinsky Random walks on hyperbolic groups
We establish a dimension formula for the harmonic measure of a finitely supported and symmetric random walk on a hyperbolic group. We also
characterize random walks for which the dimension is maximal. Our approach is based on the Green metric, a metric which provides a geometric point of view on random walks. This is a joint work with S. Blachère and P. Mathieu.
John H. Hubbard Les travaux d'Adrien Douady en géométrie analytique et en dynamique holomorphe.
L'accent sera mis sur l'influence de la personnalité d'Adrien sur les
champs qu'il a explorés.
Genadi Levin Multipliers of periodic orbits for holomorphic maps
In the beautiful theory of the Mandelbrot set by Douady and Hubbard, we single out two fragments: the uniformization of the hyperbolic components by the multiplier, and the existence of infinitely-renormalizable polynomials with non-locally connected Julia sets. In the talk, we propose a unified approach to these parts of the theory. It allows us to get further results.
Mikhail Lyubich Dynamics of unicritical polynomials
Quadratic polynomials possess some special geometric virtues that make them
tamer than their higher degree unicritical cousins. For instance, for about
15 years Yoccoz's Theorem on local connectivity of the Mandelbrot set at
non-renormalizable points had resisted to the higher degree extensions.
However, recently new analytical methods have been developed that allowed us
to bring the higher degree unicritical dynamics to the same level of
maturity as the quadratic dynamics (up to the Regular or Collet-Eckmann
Dichotomy for almost all real parameters). We will give an overview of these
results obtained jointly with Artur Avila, Jeremy Kahn and Weixiao Shen.
Curtis T. McMullen 1/7, 2/7, 4/7
We will discuss echos of surface topology and
Teichmueller theory in the dynamics of proper holomorphic
maps on the unit disk.
John Milnor Critically Periodic Cubic Polynomials
The parameter space Sp for cubic polynomial maps with a marked critical point of period p is a complicated algebraic curve whose genus increases rapidly with p. Fortunately there is a canonical procedure for choosing local coordinate charts, so that we can make local pictures of this parameter space. Each Sp consists of a compact connectedness locus together with finitely many escape region, each biholomorphic to a punctured disk. The parameter rays in the various escape regions provide a tool for studying the dynamics. This is a description of joint work with Araceli Bonifant.
Jacob Palis Open questions leading to a global perspective in dynamics
We will address one of the most challenging and central problems
in dynamical systems, meaning flows, diffeomorphisms or, more generally,
transformations, defined on a closed manifold (compact, without boundary or
an interval on the real line): can we describe the behavior in the long run
of typical trajectories for typical systems? Poincaré was probably the first to point
in this direction and stress its importance. We shall consider finite-dimensional
parameterized families of dynamics and typical will be taken in terms of Lebesgue
probability both in parameter and phase spaces. We will discuss a conjecture
stating that for a typical dynamical system, almost all trajectories have only
finitely many choices, of (transitive) attractors, where to accumulate upon in the
future. Interrelated conjectures will also be discussed.
Carsten L. Petersen Impressions of the Mandelbrot set
The Mandelbrot set M is universal in holomorphic dynamical systems via
Douady and Hubbards notion of polynomial-like mappings and quadratic-like
families. Some of the simplest non-trivial examples of quadratic-like
families are provided by perturbations within the space of quadratic
rational maps of the quadratic family Pc(z) = z2+c. These are obtained by changing the multiplier 0 of the fixed point at ∞ for Pc to
numbers λ∈ⅅ. Such a perturbation defines a holomorphic motion
Mλ of the Mandelbrot set M, naturally parametrized by
λ∈ⅅ. It turns out that S1=∂ⅅ is the natural boundary for this motion. For all boundary values |λ|=1 except λ=1
there are collisions taking place on a dense set of the boundary of
Mλ. In the talk I will survey the known results on the structure of
the radial limits or impressions of this motion of the Mandelbrot set and
non continuity along S1. Also I will pose some questions and formulate
some conjectures on the radial impressions.
Files:
Slides/f1.wmv,
Slides/f2.wmv,
Slides/f3.wmv,
Slides/f4.wmv,
Mary Rees Aeroplane Captures
In the family of quadratic rational maps, the capture construction of rational maps is an analogue of mating, which is often simpler to study. Given a critically periodic quadratic polynomial f, we denote by O(f) the full orbit of the finite critical point of f, and by On(f) the points in O(f) of preperiod n. The construction associates between 0 and 2 critically finite quadratic rational maps to each x ∈ O(f), all with one critical point of the same period as that of f, and with the other critical point in the same orbit, of preperiod n if x ∈ On(f). We call x admissible if at least one such rational map exists, and denote this rational map by gx if there is just one (which is usually the case) rather than 2.
If f is either of the two critically finite rabbit polynomials, then the map x → gx is bijective onto an easily-recognised set of critically finite rational maps in the parameter spaceV3 = {ha: a ∈ ℂ, a ≠ 0}
where
ha(z) = (z-a)(z-1)/z2,
There is a related resemblance between a part of V3, and part of the filled Julia set of the rabbit polynomial.
The situation for aeroplane captures is quite different, and, arguably, more interesting. This is related to the very different structure of one of the two parts of V3 which associate more naturally to the aeroplane polynomial. In a sideways look at this structure, we shall discuss the map x → gx, and in particular, the distribution of the numbers an,r as n → ∞, for r ≥ 1, wherean,r = #( { x ∈ On(aeroplane): #({y: gy=gx}) = r} ).
The corresponding numbers for rabbit captures are 0 for all r ≥ 2.
If f is either of the two critically finite rabbit polynomials, then the map x → gx is bijective onto an easily-recognised set of critically finite rational maps in the parameter space
The situation for aeroplane captures is quite different, and, arguably, more interesting. This is related to the very different structure of one of the two parts of V3 which associate more naturally to the aeroplane polynomial. In a sideways look at this structure, we shall discuss the map x → gx, and in particular, the distribution of the numbers an,r as n → ∞, for r ≥ 1, where
Mitsuhiro Shishikura Parabolic implosion - from discontinuity to renormalization
We consider holomorphic functions of one variable with a
parabolic periodic point (a periodic point whose derivative is a r
oot of unity.) It well-known that that such a periodic point causes
a drastic change in the dynamics when we perturb the map.
Douady and Hubbard have started an intensive study of such
a bifurcation. In particular, the Julia set can move discontinuously
and moreover for specific sequences of perturbations, the Julia set
and the dynamics on it have well-defined limit. This explains not only
the discontinuity but also a periodic structure of the parameter space
near the cusp of main cardioid in the Mandelbrot set. Key ingredient
is the analysis of Fatou coordinates and horn maps, with which
one can describe a precise form of limit dynamics (sometimes called
geometric limit). In fact, this technique lead to many interesting
results, for example, the renormalization theory for near-parabolic
fixed points and Buff-Chéritat's proof of existence of a quadratic
polynomial with Julia set of positive measure.
In the talk, I will discuss the naive idea about the phenomenon, and explain how to employ the tools due to Douady and Hubbard, to show the discontinuity etc. If the times allows, I will try to cover more recent development of the theory.
In the talk, I will discuss the naive idea about the phenomenon, and explain how to employ the tools due to Douady and Hubbard, to show the discontinuity etc. If the times allows, I will try to cover more recent development of the theory.
Dennis Sullivan String configurations in dimensions 2,3,4,...
After removing a subset of infinite codimension the free loop space
of a manifold is naturally stratified. The strata are labeled by even
valence symmetric graphs called string configurations. The finite
type invariants of Vassiliev in dimension three relate to this
picture.
Adding a cyclic structure to the graphs brings in cell decompositions
of moduli spaces of riemann surfaces and the operations of string
topology.
Sebastian van Strien On the interface of real and holomorphic dynamics
In the talk I will discuss three results:
(1) complex bounds and rigidity for non-renormalizable polynomials with only hyperbolic periodic points (joint work with Kozlovski)
(2) monotonicity of entropy for real polynomials (joint work with Bruin and Shen)
(3) description of rational maps for which multipliers of periodic points are always real.
(1) complex bounds and rigidity for non-renormalizable polynomials with only hyperbolic periodic points (joint work with Kozlovski)
(2) monotonicity of entropy for real polynomials (joint work with Bruin and Shen)
(3) description of rational maps for which multipliers of periodic points are always real.
Alberto Verjovsky Hedlund´s theorem on minimality of horocycle flows for Riemann surface laminations
In this talk we indicate a proof of the dichotomy: either the
horocycle flow on the unit tangent bundle of a compact lamination by surfaces
of strictly negative curvature is minimal or else the lamination is given by
the orbits of a locally-free action of the affine group
Michel Zinsmeister Branner-Hubbard Motion without Dynamics
In one of his latest work, Adrien Douady noticed that
Branner-Hubbard Motion can be defined for all Dirichlet-regular compact sets
of the plane, but the lack of dynamics makes the generalisation of the
classical results problematic. We will survey about Douady's results on
this, state and explain his conjectures and give some applications of the
theory.
