Hamiltonian pseudo-representations

Abstract

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 normed Lie algebra, it converges to 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 Gromov-Eliashberg Theorem, it is to say that the group of symplectic diffeomorphisms is C^0-closed in the group of all diffeomorphisms.

16 pages. Main result extended to a large class of Lie algebras

References in corpus (3)

• Commuting Hamiltonians and multi-time Hamilton-Jacobi equations
• C^0-rigidity of Poisson brackets
• The 2/3 - convergence rate for the Poisson bracket

Cited by in corpus (5)

• New energy-capacity-type inequalities and uniqueness of continuous Hamiltonians
• Poisson brackets, quasi-states and symplectic integrators
• Weakly Hamiltonian actions
• Symplectic rigidity and weak commutativity
• $C^0$-rigidity of Poisson diffeomorphisms