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 3-geometries (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 3-manifold" 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 step in a __program of Yoccoz__ (first and last lecture at Collège de France 1995-2016) in the case of the dynamics of *Hénon-like maps* __ by giving__ a topological and combinatorial definition for many of those which display a statistical behavior. It is a generalization of the Benedicks-Carleson 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 Mañe-Sad-Sullivan, 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 Aoki-Moriyasu-Sumi.