Heavy subsets and non-contractible trajectories

Abstract

Biran, Polterovich and Salamon defined a relative symplectic capacity which indicates the existence of 1-periodic non-contractible closed trajectories of Hamiltonian isotopies. Many of researches have used the Hamiltonian Floer theory on non-contractible trajectories for giving upper bounds of Biran-Polterovich-Salamon's capacities. However, in the present paper, we use the Oh-Schwarz spectral invariants which are defined in terms of the Hamiltonian Floer theory on contractible trajectories for a similar purpose.

Floer homology in the cotangent bundle of a closed Finsler manifold and noncontractible periodic orbits

“…On the other hand, there are some other results [20,42] on the existence of periodic orbits on cotangent bundle which do not fit into the general Riemannian framework of [41]. In particular, these results have important applications including the preservation of marked length spectrum under symplectomorphisms, the existence of noncontractible periodic orbits for Hamiltonian Lorentzian systems, etc.…”

“…(a) Generalizing theorem 2 of [42] to the Lie group setting (theorem 1.8), (b) Preservation of minimal Finsler length of closed geodesics in any given free homotopy class by symplectomorphisms (theorem 1.5), (c) Existence of periodic orbits for Hamiltonian systems separating two Lagrangian submanifolds (theorem 1.11), (d) Existence of periodic orbits for Hamiltonians on noncompact domains (theorem 1.12), (e) Existence of periodic orbits for Lorentzian Hamiltonian in higher dimensional case (theorem 1.13), (f ) Partial solution to a conjecture of Kawasaki in [20] (theorem 1.15), (g) Results on squeezing/nonsqueezing theorem on torus cotangent bundles (theorems 1. 16-1.18).…”

Floer homology in the cotangent bundle of a closed Finsler manifold and noncontractible periodic orbits

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We show the the existence of noncontractible periodic orbits for every compactly supported time-dependent Hamiltonian on the open unit disk cotangent bundle of a Finsler manifold provided that the Hamiltonian is sufficiently large over the zero section. This result solves a conjecture of Irie [19] and generalizes the previous results [8,40,42] etc.We then obtain a number of applications including: (1) preservation of Finsler lengths of closed geodesics by symplectomorphisms, (2) existence of periodic orbits for Hamiltonian systems separating two Lagrangian submanifolds, (3) existence of periodic orbits for Hamiltonians on noncompact domains, (4) existence of periodic orbits for Lorentzian Hamiltonian in higher dimensional case, (5) partial solution to a conjecture of Kawasaki in [22], (6) results on squeezing/nonsqueezing theorem on torus cotangent bundles.

…”

“…On the other hand, there are some other results [42,22] on the existence of periodic orbits on cotangent bundle which does not fit into the general Riemannian framework of [40]. In particular, these results have important applications including the existence of noncontractible periodic orbits for Hamiltonian Lorentzian systems, etc.…”

“…(1) preservation of minimal Finsler lengths of closed geodesics by symplectomorphisms (Corollary 1.5), (2) existence of periodic orbits for Hamiltonian systems separating two Lagrangian submanifolds (Theorem 1.7), (3) existence of periodic orbits for Hamiltonians on noncompact domains (Theorem 1.8), (4) existence of periodic orbits for Lorentzian Hamiltonian in higher dimensional case (Theorem 1.9), (5) partial solution to a conjecture of Kawasaki in [22] (Theorem 1.11), (6) results on squeezing/nonsqueezing theorem on torus cotangent bundles (Theorem 1.12, 1.13, 1.14).…”