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

2017/01/01 by Morimichi Kawasaki, Ryuma Orita
Computer Science · Mathematics · #Combinatorics #Computation #Contractible space #Geometric and Algebraic Topology #Geometry #Hamiltonian (control theory) #Hamiltonian system #Lagrangian #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Periodic orbits #Pure mathematics #Submanifold #Symplectic geometry #Symplectic manifold #Torus #Upper and lower bounds #math.DS #math.SG #msc:37J10 #msc:37J45 #msc:53D40 #semigroups and automata theory

paper · pdf · doi:10.3934/jmd.2017013

published as J. Mod. Dyn. 11 (2017) 313-339 · 24 pages, 6 figures

openalex publication_date 2017/01/01 · arxiv created 2017/03/06 · arxiv updated 2017/04/25 · openalex created_date 2017/04/28 · openalex updated_date 2026/08/05

Abstract

The first author [9] introduced a relative symplectic capacity C for a symplectic manifold (N,ω_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)×ℝ/ℤ. In the present paper, we give an exact computation of the capacity C of the 2n-torus \mathbbT²ⁿ relative to a Lagrangian submanifold \mathbbTⁿ which implies the existence of non-contractible Hamiltonian periodic trajectories on A_R×\mathbbT²ⁿ. Moreover, we give a lower bound on the number of such trajectories.

Citations

• Noncontractible periodic orbits in cotangent bundles and Floer homology
• Symplectic fixed points and holomorphic spheres
• Existence of noncontractible periodic orbits of Hamiltonian system separating two Lagrangian tori on T^*\Tⁿ with application to non convex Hamiltonian systems
• Morse theory for periodic solutions of hamiltonian systems and the maslov index
• Functors and Computations in Floer homology with Applications Part II