We construct a quotient Aubry set isometric to the unit interval. We also construct several related examples of Aubry sets.

We survey some recently obtained generic consequences of the existence of homoclinic tangencies in diffeomorphisms of surfaces. Among other things, it has been shown that they give rise to invariant topologically transitive sets with maximal Hausdorff dimension, that they prohibit the existence of various kinds of symbolic extensions and that they form an impediment to the existence of Sinai–Ruelle–Bowen (SRB) measures. The main new result described here, together with a positive answer to an as yet unproved conjecture of Palis, would prove that generically on surfaces, SRB measures only exist on uniformly hyperbolic attractors.

The so-called Herman–Światek theorem is generalized to holomorphic self-homeomorphisms of quasi-circles. This result implies an unpublished theorem of Michel Herman: If a Siegel disk or Arnold–Herman ring for a rational map has a boundary component, which is a quasi-circle containing a critical point, then the associated rotation number is Diophantine of exponent 2.

We consider Gevrey perturbations H of a completely integrable non-degenerate Gevrey Hamiltonian H0. Given a Cantor set $\Omega_\kappa$ defined by a Diophantine condition, we find a family of Kolmogorov–Arnold–Moser (KAM) invariant tori of H with frequencies $\omega\in \Omega_\kappa$ which is Gevrey smooth in a Whitney sense. Moreover, we obtain a symplectic Gevrey normal form of the Hamiltonian in a neighborhood of the union $\Lambda$ of the invariant tori. This leads to effective stability of the quasi-periodic motion near $\Lambda$.

We study an analytic Hamiltonian system with two degrees of freedom, having the origin as an elliptic singularity. We assume that the full Birkhoff normal form exists and is divisible by its quadratic part, being indefinite. We show that under the Bruno condition and under the restriction to the zero energy surface, a real analytic transformation into a normal form exists. Such a normal form coincides with the restriction of the Birkhoff normal form to the zero energy surface up to an order as large as we want.

Nous nous intéressons aux structures de Poisson analytiques singulières en un point et de partie linéaire non-nulle en ce point. En utilisant des travaux récents de l'auteur sur la normalisation holomorphe de familles commutatives de champs de vecteurs holomorphes singuliers, nous donnons des résultats de normalisation holomorphe de certaines structures de Poisson.