C^0-rigidity of Poisson brackets

Abstract

Consider a functional associating to a pair of compactly supported smooth functions on a symplectic manifold the maximum of their Poisson bracket. We show that this functional is lower semi-continuous with respect to the product uniform (C^0) norm on the space of pairs of such functions. This extends previous results of Cardin-Viterbo and Zapolsky. The proof involves theory of geodesics of the Hofer metric on the group of Hamiltonian diffeomorphisms. We also discuss a failure of a similar semi-continuity phenomenon for multiple Poisson brackets of three or more functions.

Latex, 11 pages

References in corpus (2)

• Commuting Hamiltonians and multi-time Hamilton-Jacobi equations
• Minimal atlases of closed symplectic manifolds

Cited by in corpus (11)

• Quasi-morphisms on contactomorphism groups and contact rigidity
• Quasi-morphisms and symplectic quasi-states for convex symplectic manifolds
• Quasi-morphisms and quasi-states in symplectic topology
• Hamiltonian pseudo-representations
• Observations on the Hofer distance between closed subsets
• New energy-capacity-type inequalities and uniqueness of continuous Hamiltonians
• The 2/3 - convergence rate for the Poisson bracket
• C^0-rigidity of the double Poisson bracket
• Poisson brackets, quasi-states and symplectic integrators
• Monodromy in Hamiltonian Floer theory
• Poisson brackets and symplectic invariants