Heavy subsets and non-contractible trajectories

Abstract

Entov and Polterovich defined heaviness for closed subsets of a symplectic manifold by using the Hamiltonian Floer theory on contractible trajectories. Heavy subsets are known to be non-displaceable. In the present paper, we define a relative symplectic capacity $C(M,X,R;e)$ for a symplectic manifold $(M,\omega )$ and its subset $X$ which measures the existence of non-contractible trajectories of Hamiltonian isotopies on the product with annulus. We prove that $C(M,X,R;e)$ is finite if $(M,\omega )$ is monotone and $X$ is a heavy subset. We also prove that $C(M,X,R;e)$ is infinite if $X$ is a displaceable compact subset.

changed the constituion of the paper very much, added new section (Section 5 and 7), the inequality in the main theorem contained a mistake and fixed it, fixed many other small mistakes, 20 pages

References in corpus (2)

• Non-contractible Periodic Orbits in Hamiltonian Dynamics on Closed Symplectic Manifolds
• Non-contractible periodic orbits, Gromov invariants, and Floer-theoretic torsions

Cited by in corpus (3)

• Computation of annular capacity by Hamiltonian Floer theory of non-contractible periodic trajectories
• Function theoretical applications of Lagrangian spectral invariants
• Non-contractible orbits for Hamiltonian functions on Riemann surfaces