
Pierre Berger
Chargé de recherche CNRS en Systèmes Dynamiques Laboratoire Analyse, Géométrie & Applications
pierre.berger((at))normalesup.org 99 avenue J.B. Clément  93430 Villetaneuse  France 

I am the coordinator of many exhibitions on 3 manifolds, showing images, installations and movies made alone or in collaboration with scientists and artists. The pieces are based on new researchs in pure mathematics. For instance, here I solved problems from Thurston's school at the Geometry Center in the 90's, by depicting the innner views in 7 over 8 of 3geometries (among which 3 were kwnon before). Thousands of teenagers have seen these works, in the Paris suburbs, Rio de Janeiro, and soon as a permanent exhibition in Mexico. During the 7ECM at Berlin, our exhibition "view in 3manifold" had occupied an important place.
My inital aim is to understand how can we understand the trajectories given by a differentiale equation, or similarly the iterations of a diffeomorphisms, from the topological, combinatorial and statistical viewpoints.
In the positive direction, I solved a program of Yoccoz (first and last lecture at Collège de France 19952016) in the case of the dynamics of Hénonlike maps by giving a topological and combinatorial definition for many of those which display a statistical behavior. It is a generalization of the BenedicksCarleson Theorem. The Hénon like maps are C^{2} perturbations of the planar map:
I solved a question by Carleson by showing that every maximal Lyapunov exponent of each of these maps is uniformly positive. Recall that given a surface map f, the maximal Lyapunov exponent of f at a point z is limsup (1/n) log D_{z}f^{n}.
Recently with D. Turaev, we proved the positive entropy conjecture of Herman (ICM 1998) by showing that any C^{∞} area preserving, surface diffeomorphism which displays an eliptic periodic point, can be C^{∞}approximated by one which displays a positve metric entropy (the maximal Lyapunov exponents are non zero for a set of points of positive Lebesgue measure).
In the negative direction, I broke a few conjectures from the 90's by showing that a typical diffeomorphisms  in the sense of Kolmogorov  can display infinitely many attractors. I hope it is the first step to show that the physical behavior of some typical dynamics cannot be described by means of statistics, as claimed by my program on emergence.
Also I gave a negative answer to a problem of Arnold (1992) by showing that a typical diffeomorphism in the sense of Kolmogorov can display a super exponential growth of the number of periodic points.
Works in Strutural stability. This direction asks in which extend the structure of the orbits of a dynamical systems is the same as its perturbations. The strongest form asks for a conjugacy, that is a homeomorphism between the sets orbits of the one other. A dynamics satisfying such a property is called structurally stable. A main problem in dynamcal system is to describe in a satisfactory way those dynamics. This problem compasses some of the most famous conjectures and theorems in dynamical systems, such as the Fatou conjecture for quadratic maps, the Lambda lemma for rational functions of Lyubich and MañeSadSullivan, the strutural stability conjecture by Smale and its solution for C^{ 1}diffeomorphisms, by Robin, Robinson and Mañé.
The Lambda lemma describes the structurally stable dynamics as those which have robustely the following property: the periodic points are all hyperbolic. With R. Dujardin we showed a probabilistic version of the Lambda Lemma, for complex automorphisms of C^{2}.
The structural stability conjecture and the Fatou conjecture aim to describe the structurally stable dynamics as those which are uniformly hyperbolic (in particular all the periodic points are hyperbolic and this uniformly). With A. Rovella, we formulate a version of the strutural stability conjecture for C^{1}non invertible maps, and with A. Kocsard, we showed that one direction of this conjecture holds true, the other direction of this conjecture is known in many cases by the works of Mañé and AokiMoriyasuSumi.