Hamiltonian pseudo-representations

Vincent Humilière (1)

1 CMLS - Centre de Mathématiques Laurent Schwartz

The question studied here is the behavior of the Poisson bracket under C^0-perturbations. In this purpose, we introduce the notion of pseudo-representation and prove that for a nilpotent Lie algebra, it converges to a representation. This question remains open for general Lie algebras. An unexpected consequence of this result is that for many non-closed symplectic manifolds (including cotangent bundles), the group of Hamiltonian diffeomorphisms (with no assumptions on supports) has no C^{-1} bi-invariant metric. Our methods also provide a new proof of Gromov-Eliashberg Theorem, it is to say that the group of symplectic diffeomorphisms is C^0-closed in the group of all diffeomorphisms.

Mots clés: Poisson brackets; biinvariant metrics; C^0 symplectic topology

Domaines: Géométrie symplectique [math.SG]; Physique mathématique [math-ph]

Vincent Humilière. Hamiltonian pseudo-representations. 2007. ⟨hal-00136107v1⟩