We have obtained results towards a global panorama of differentiable dynamical systems, and in particular
- a proof of Palis conjecture about C1 flows far from homoclinic tangencies,
- a partial answer to Bonatti’s conjecture about the finiteness of the number of chain-recurrence classes for C1-generic diffeomorphisms far from homoclinic tangencies,
- a characterization of the class of discretized Anosov flows,
- a study of multi-singular hyperbolic flows in higher dimension.
In the setting of surface dynamics, we have developed tools allowing to address critical behaviors:
- introduction of the class of mildly dissipative dynamics in order to renormalize Hénon maps with zero entropy,
- interaction between homoclinic tangencies and heterodimensional cycles, producing germ-typical Newhouse phenomenon.
The fine structure and properties of the non-uniformly hyperbolic locus have been studied, through codings and Yomdin’s technique. This has lead to results on:
- finiteness of homoclinic classes and measures maximizing the entropy,
- continuity of Lyapunov exponents, and introduction of a new class of non-uniform hyperbolicity,
- quantitative statistical properties of equilibrium measures.
Results on physical and geometrical measures have been obtained:
- existence and rigidity of u-Gibbs states in the partially hyperbolic setting,
- existence of SRB measures for surface diffeomorphisms (Viana’s conjecture),
- invariance principle and criterion for non-vanishing Lyapunov exponents of partially hyperbolic systems with non-compact center bundle,
- stable ergodicity of generic conservative partially-hyperbolic systems.
Some interactions with other dynamical settings have also been explored (geodesic flows, billiards, interval exchange transformations).
This has lead to about 50 research papers and the organization of 8 meetings.
