Images: Here, I solved problems from Thurston's school at the Geometry Center in the 90s, by simulating the first inner views in most 3-geometries (only 3 were simulated before).
Sounds: With Jimena Royo-Leteyer, we gave the first acoustic simulation in several 3-manifolds, see this online experimentation.
Exibitions These experiment were part of many exhibitions on 3-manifolds that I realized, showing images, installations, and movies created either alone or in collaboration with scientists and artists. These exhibitions took place at Paris suburbs, Rio de Janeiro, permanently at UNAM museum of Mexico, at prominent place at the 7ECM (Berlin), and during more than 3 months at the IHP in Paris, see these exhibitions or the esthétopies exhibition.
I am currently finishing the conception and production of a new exhibition on wild dynamics.
My initial aim is to understand how we can comprehend the trajectories given by a differential equation or similarly the iterations of a diffeomorphism, from the topological, combinatorial, and statistical viewpoints.
Works in analytic symplectic dynamics: I showed the existence of an analytic symplectomorphism of the cylinder or sphere with zero or only two periodic points, which is not conjugated to a rotation. This disproved two conjectures by Birkhoff (1941). Additionally, a counter-example can be chosen to exhibit statistical behavior requiring a super-exponential number of probability measures (its local emergence has maximal order). Another example constructed on the cylinder is ergodic. This solves a question of Herman (ICM 1998). Using method from complex geometry, I then generalized this result for other surfaces: the disk and the sphere. As corollary I obtained a proof of a conjecture by Birkhoff 1927 (existence of unstable elliptic point for system with two degree of freedom). This also solved problems by Katok-Fayad and Herman.
Works in renormalization theory: With Gourmelon and Helfter, we showed that any dynamics can be obtained as a perturbation of the identity up to renormalization (rescaling space and time). This solved a problem posed by Ruelle and Takens (1971).
Works in bifurcation theory: I disproved several conjectures from the 90s by showing that a typical diffeomorphism—in the sense of Kolmogorov—can display infinitely many attractors. I hope this is the first step in showing that the physical behavior of some typical dynamics is too complex to be described through classical statistical methods. This is the subject of my program on emergence which is supported by the ERC since 2019.
I provided a negative answer to a problem posed by Arnold (1992) by showing that a typical diffeomorphism, in the sense of Kolmogorov, can exhibit a super-exponential growth in the number of periodic points.
Works in holomorphic dynamics: With S. Biebler, we showed the existence of a real polynomial automorphism of the plane that not only displays a wandering real and complex Fatou component (solving a problem of Milnor and Bedford-Smillie from the 90s) but also has a statistical behavior that requires an exponential number of probability measures to describe (its emergence has positive order). We also demonstrated that this behavior is locally dense in the Newhouse domain.
The Lambda Lemma characterizes structurally stable dynamics as those that exhibit only hyperbolic periodic points and this robustly. With R. Dujardin, we showed a probabilistic version of the Lambda Lemma for complex automorphisms of C2.
Works on attractors and positive metric entropy:, I solved a step in a program of Yoccoz (first and last lecture at Collège de France, 1995-2016) in the context 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.
I solved a question posed by Carleson by showing that every maximal Lyapunov exponent of these maps is uniformly positive. Recall that given a surface map f, the maximal Lyapunov exponent of f at a point z is lim sup (1/n) log ||Dzfn||.
With D. Turaev, we proved Herman's positive entropy conjecture (ICM 1998) by showing that any C∞ area-preserving surface diffeomorphism with an elliptic periodic point can be C∞-approximated by one with positive metric entropy (where the maximal Lyapunov exponents are non-zero for a set of points with positive Lebesgue measure).
Works in Structural Stability: This direction seeks to determine the extent to which the structure of a dynamical system's orbits remains unchanged under perturbations. The strongest form of this property asks for a conjugacy, i.e., a homeomorphism between the sets of orbits. A dynamical system that satisfies such a property is called structurally stable. One of the main problems in dynamical systems is to provide a comprehensive description of such systems.
The structural stability conjecture and the Fatou conjecture propose that structurally stable dynamics are uniformly hyperbolic (i.e., all periodic points are uniformly hyperbolic). With A. Rovella, we formulated a version of the structural stability conjecture for C1 non-invertible maps. Additionally, with A. Kocsard, we showed that one direction of this conjecture holds true, while the other has been proven in many cases by the works of Mañé and Aoki-Moriyasu-Sumi.