Computation of annular capacity by Hamiltonian Floer theory of non-contractible periodic trajectories

Morimichi Kawasaki, Ryuma Orita
Journal of Modern Dynamics, vol. 11 (2017), pp. 313–339
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Overview

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For a Hamiltonian system on the product of an annulus and a symplectic manifold, one can ask what forces a periodic trajectory with a prescribed winding around the annulus. The relative symplectic capacity $C$, introduced by Kawasaki, measures the threshold on the Hamiltonian along a chosen subset that guarantees such a trajectory.

This paper computes $C$ exactly for the standard symplectic torus $\mathbb{T}^{2n}$ relative to its standard Lagrangian torus $\mathbb{T}^n$. The calculation gives a sharp existence condition for non-contractible periodic trajectories on $A_R\times\mathbb{T}^{2n}$, where $A_R=(-R,R)\times\mathbb{R}/\mathbb{Z}$. Under a non-degeneracy assumption, it also gives a lower bound on their number. The proof uses Hamiltonian Floer theory for non-contractible trajectories.

@article{kawasaki2017computation,
  author        = {Kawasaki, Morimichi and Orita, Ryuma},
  title         = {{Computation of annular capacity by Hamiltonian Floer theory of non-contractible periodic trajectories}},
  journal       = {Journal of Modern Dynamics},
  volume        = {11},
  pages         = {313--339},
  year          = {2017},
  doi           = {10.3934/jmd.2017013},
  eprint        = {1703.01730},
  archivePrefix = {arXiv}
}