New energy-capacity-type inequalities and uniqueness of continuous Hamiltonians

Abstract

We prove a new variant of the energy-capacity inequality for closed rational symplectic manifolds (as well as certain open manifolds such as ${\mathbb{R}}^{2n}$, cotangent bundle of closed manifolds…) and we derive some consequences to $C^0$–symplectic topology. Namely, we prove that a continuous function which is a uniform limit of smooth normalized Hamiltonians whose flows converge to the identity for the spectral (or Hofer’s) distance must vanish. This gives a new proof of uniqueness of continuous generating Hamiltonian for hameomorphisms. This also allows us to improve a result by Cardin and Viterbo on the $C^0$–rigidity of the Poisson bracket.