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 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 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.

18 pages. v2. Several minor changes. Reference list updated. To appear in Commentarii Mathematici Helvetici

References in corpus (7)

• On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows
• Coisotropic rigidity and C^0-symplectic geometry
• Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds
• C^0-rigidity of Poisson brackets
• Quasi-morphisms and symplectic quasi-states for convex symplectic manifolds
• Descent and C^0-rigidity of spectral invariants on monotone symplectic manifolds
• Poisson brackets, quasi-states and symplectic integrators

Cited by in corpus (5)

• Coisotropic rigidity and C^0-symplectic geometry
• Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces
• A C^0 counterexample to the Arnold conjecture
• Unboundedness of the Lagrangian Hofer distance in the Euclidean ball
• Some properties of Hamiltonian homeomorphisms on closed aspherical surfaces