Hamiltonian pseudo-representations

Abstract

The question studied here is the behavior of the Poisson bracket under C0-perturbations. For this purpose we introduce the notion of pseudo-representation and prove that the limit of a converging pseudo-representation of any normed Lie algebra is a representation.

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 the Gromov