Journal of Modern Dynamics (JMD)

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

Pages: 313 - 339, Volume 11, 2017      doi:10.3934/jmd.2017013

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Morimichi Kawasaki - Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang 790-784, Republic of Korea (email)
Ryuma Orita - Graduate School of Mathematical Sciences, University of Tokyo, Tokyo 153-0041, Japan (email)

Abstract: The first author [9] introduced a relative symplectic capacity $C$ for a symplectic manifold $(N,\omega_N)$ and its subset $X$ which measures the existence of non-contractible periodic trajectories of Hamiltonian isotopies on the product of $N$ with the annulus $A_R=(-R,R)\times\mathbb{R}/\mathbb{Z}$. In the present paper, we give an exact computation of the capacity $C$ of the $2n$-torus $\mathbb{T}^{2n}$ relative to a Lagrangian submanifold $\mathbb{T}^n$ which implies the existence of non-contractible Hamiltonian periodic trajectories on $A_R\times\mathbb{T}^{2n}$. Moreover, we give a lower bound on the number of such trajectories.

Keywords:  Hamiltonian Floer theory, periodic trajectory of Hamiltonian isotopy, Biran–Polterovich–Salamon capacity.
Mathematics Subject Classification:  Primary: 53D40, 37J10; Secondary: 37J45.

Received: May 2016;      Revised: February 2017;      Available Online: April 2017.