Hamiltonian pseudo-representations

Abstract

The question studied here is the behavior of the Poisson bracket under $C^0$-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–Eliashberg Theorem, which says that the group of symplectic diffeomorphisms is $C^0$-closed in the group of all diffeomorphisms.