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

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

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 4, 2016 Revised  February 14, 2017 Published  April 2017

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.

Citation: Morimichi Kawasaki, Ryuma Orita. Computation of annular capacity by Hamiltonian Floer theory of non-contractible periodic trajectories. Journal of Modern Dynamics, 2017, 11: 313-339. doi: 10.3934/jmd.2017013
References:
[1]

P. Biran, L. Polterovich, D. Salamon, Propagation in Hamiltonian dynamics and relative symplectic homology, Duke Math.J., 119 (2003), 65-118.

[2]

K. Cieliebak, Handle attaching in symplectic homology and the chord conjecture, J. Eur.Math.Soc.(JEMS), 4 (2002), 115-142.

[3]

K. Cieliebak, A. Floer, H. Hofer, Symplectic homology.Ⅱ.A general construction, Math.Zeit., 218 (1995), 103-122.

[4]

A. Floer, Symplectic fixed points and holomorphic spheres, Comm.Math.Phys., 120 (1989), 575-611.

[5]

A. Floer, H. Hofer, Symplectic homology.Ⅰ.Open sets in $\mathbb{C}^n$, Math.Zeit., 215 (1994), 37-88.

[6]

A. Floer, H. Hofer, D. Salamon, Transversality in elliptic Morse theory for the symplectic action, Duke Math.J., 80 (1995), 251-292.

[7]

U. Frauenfelder, F. Schlenk, Hamiltonian dynamics on convex symplectic manifolds, Israel J.Math., 159 (2007), 1-56.

[8]

H. Ishiguro, Non-contractible orbits for Hamiltonian functions on Riemann surfaces, arXiv: 1612.07062, (2016).

[9]

M. Kawasaki, Heavy subsets and non-contractible trajectories, arXiv: 1606.01964, (2016).

[10]

C. Niche, Non-contractible periodic orbits of Hamiltonian flows on twisted cotangent bundles, Discrete Contin.Dyn.Syst., 14 (2006), 617-630.

[11]

M. Poźniak, Floer homology, Novikov rings and clean intersections, in Northern California Symplectic Geometry Seminar, (eds. Y. Eliashberg, D. Fuchs, T. Ratiu, and A. Weinstein), Amer. Math. Soc. Transl. Ser. 2 196, Amer. Math. Soc. , Providence, 1999,119-181.

[12]

D. Salamon, Lectures on Floer homology, in Symplectic Geometry and Topology (Park City, Utah, 1997), IAS/Park City Math. Ser. 7, Amer. Math. Soc. , Providence, 1999,143-230.

[13]

D. Salamon, E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm.Pure Appl.Math., 45 (1992), 1303-1360.

[14]

M. Usher, The sharp energy-capacity inequality, Commun.Contemp.Math., 12 (2010), 457-473.

[15]

C. Viterbo, Functors and computations in Floer homology with applications Ⅰ, Geom.funct.anal., 9 (1999), 985-1033.

[16]

J. Weber, Noncontractible periodic orbits in cotangent bundles and Floer homology, Duke Math.J., 133 (2006), 527-568.

[17]

J. Xue, Existence of noncontractible periodic orbits of Hamiltonian system separating two Lagrangian tori on $T^{\ast}\mathbb{T}^{n}$ with application to non convex Hamiltonian systems, to appear in J. Symplectic Geom. , arXiv: 1408.5193, (2014).

show all references

References:
[1]

P. Biran, L. Polterovich, D. Salamon, Propagation in Hamiltonian dynamics and relative symplectic homology, Duke Math.J., 119 (2003), 65-118.

[2]

K. Cieliebak, Handle attaching in symplectic homology and the chord conjecture, J. Eur.Math.Soc.(JEMS), 4 (2002), 115-142.

[3]

K. Cieliebak, A. Floer, H. Hofer, Symplectic homology.Ⅱ.A general construction, Math.Zeit., 218 (1995), 103-122.

[4]

A. Floer, Symplectic fixed points and holomorphic spheres, Comm.Math.Phys., 120 (1989), 575-611.

[5]

A. Floer, H. Hofer, Symplectic homology.Ⅰ.Open sets in $\mathbb{C}^n$, Math.Zeit., 215 (1994), 37-88.

[6]

A. Floer, H. Hofer, D. Salamon, Transversality in elliptic Morse theory for the symplectic action, Duke Math.J., 80 (1995), 251-292.

[7]

U. Frauenfelder, F. Schlenk, Hamiltonian dynamics on convex symplectic manifolds, Israel J.Math., 159 (2007), 1-56.

[8]

H. Ishiguro, Non-contractible orbits for Hamiltonian functions on Riemann surfaces, arXiv: 1612.07062, (2016).

[9]

M. Kawasaki, Heavy subsets and non-contractible trajectories, arXiv: 1606.01964, (2016).

[10]

C. Niche, Non-contractible periodic orbits of Hamiltonian flows on twisted cotangent bundles, Discrete Contin.Dyn.Syst., 14 (2006), 617-630.

[11]

M. Poźniak, Floer homology, Novikov rings and clean intersections, in Northern California Symplectic Geometry Seminar, (eds. Y. Eliashberg, D. Fuchs, T. Ratiu, and A. Weinstein), Amer. Math. Soc. Transl. Ser. 2 196, Amer. Math. Soc. , Providence, 1999,119-181.

[12]

D. Salamon, Lectures on Floer homology, in Symplectic Geometry and Topology (Park City, Utah, 1997), IAS/Park City Math. Ser. 7, Amer. Math. Soc. , Providence, 1999,143-230.

[13]

D. Salamon, E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm.Pure Appl.Math., 45 (1992), 1303-1360.

[14]

M. Usher, The sharp energy-capacity inequality, Commun.Contemp.Math., 12 (2010), 457-473.

[15]

C. Viterbo, Functors and computations in Floer homology with applications Ⅰ, Geom.funct.anal., 9 (1999), 985-1033.

[16]

J. Weber, Noncontractible periodic orbits in cotangent bundles and Floer homology, Duke Math.J., 133 (2006), 527-568.

[17]

J. Xue, Existence of noncontractible periodic orbits of Hamiltonian system separating two Lagrangian tori on $T^{\ast}\mathbb{T}^{n}$ with application to non convex Hamiltonian systems, to appear in J. Symplectic Geom. , arXiv: 1408.5193, (2014).

Figure 1.  Outline of the graphs of $f_s$ (for $s\geq 1$ and $s\leq -1$)
Figure 2.  Outline of the graphs of $H_s$ (for $s\geq 1$ and $s\leq -1$) in the case $n=1$
Figure 3.  Outline of the graph of $H_s$ (for $s\geq 1$) in the direction of $p_0$
Figure 4.  Outline of the graph of $H_s$ (for $s\leq -1$) in the direction of $p_0$
Figure 5.  Outline of the graph of $H_T\natural(\varepsilon \rho_TF_T)$ in the case $n=1$
Figure 6.  Outline of the graph of $\widetilde{H}$ in the case $n=1$
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