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

Morimichi Kawasaki; Ryuma Orita

doi:10.3934/jmd.2017013

Article Contents

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

This volume Previous Article Exponential mixing and smooth classification of commuting expanding maps Next Article Normal forms for non-uniform contractions

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

Morimichi Kawasaki^1, and
Ryuma Orita^2,

1.
Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang 790-784, Republic of Korea
2.
Graduate School of Mathematical Sciences, University of Tokyo, Tokyo 153-0041, Japan

Received: May 03, 2016

Revised: February 13, 2017

Published: March 2017

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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.

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Figure 1. Outline of the graphs of $f_s$ (for $s\geq 1$ and $s\leq -1$)

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Figure 2. Outline of the graphs of $H_s$ (for $s\geq 1$ and $s\leq -1$) in the case $n=1$

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Figure 3. Outline of the graph of $H_s$ (for $s\geq 1$) in the direction of $p_0$

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Figure 4. Outline of the graph of $H_s$ (for $s\leq -1$) in the direction of $p_0$

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Figure 5. Outline of the graph of $H_T\natural(\varepsilon \rho_TF_T)$ in the case $n=1$

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Figure 6. Outline of the graph of $\widetilde{H}$ in the case $n=1$

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References

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