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 research 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 known before). Thousands of visitor have seen these works, in the Paris suburbs, Rio de Janeiro, and in a permanent exhibition in Mexico. During the 7ECM at Berlin, our exhibition "view in 3-manifold" had occupied an important place. My exhibition esthétopies stayed more than 3 months at IHP, in Paris center during 2017.
My initial aim is to understand how can we understand the trajectories given by a differential equation, or similarly the iterations of a diffeomorphism, 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 C2 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 ||Dzfn||.
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 elliptic periodic point, can be C∞-approximated by one which displays a positive 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.
With S. Biebler we showed the existence of a real polynomial automorphism of the plane which displays not only a wandering real and complex Fatou component (solving a problem of Milnor and Bedford-Smillie from 90), but also whose statistic behavior needs a straight exponential number of probability measures to be described (its emergence has positive order). We also proved that such behavior is locally dense in the Newhouse domain of surface diffeomorphism (solving a the so-called last Takens' problem) in any topology (smooth and analytic).
Recently, I showed the existence of an analytic symplectomorphism of the cylinder or the sphere with zero or only 2 periodic points, which is not conjugated to a rotation. This disproved two conjectures of Birkhoff (1941). Moreover a counter example can be chosen to have a statistic behavior which needs a super exponential number of probability measures to be described (its local emergence has maximal order). Another example constructed on the cylinder is ergodic. This solves a question of Herman (1998).
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 dynamical 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 structural 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 robustly 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 C2.
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 structural stability conjecture for C1-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.