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

Abstract

We show that the existence of noncontractible periodic orbits for compactly supported time-dependent Hamiltonian on the disk cotangent bundle of a Finsler manifold provided that the Hamiltonian is sufficiently large over the zero section. We generalize the BPS capacities and earlier constructions of Weber (2006 Duke Math. J. 133, 527-568) and other authors Biran et al (2003 Duke Math. J. 119, 65-118) to the Finsler setting. We then obtain a number of applications including: (1) generalizing the main theorem of Xue (2017 J. Symplectic Geom. 15, 905-936) to the Lie group setting, (2) preservation of minimal Finsler length of closed geodesics in any given free homotopy class by symplectomorphisms, (3) existence of periodic orbits for Hamiltonian systems separating two Lagrangian submanifolds, (4) existence of periodic orbits for Hamiltonians on noncompact domains, (5) existence of periodic orbits for Lorentzian Hamiltonian in higher dimensional case, (6) partial solution to a conjecture of Kawasaki (2016 Heavy subsets and non-contractible trajectories (arXiv:1606.01964)), (7) results on squeezing/nonsqueezing theorem on torus cotangent bundles, etc.

arXiv comment51 pages, 9 figures; published version; v.3 includes some minor improvements to the exposition suggested by the referees

References in corpus (7)

• Floer homology and the heat flow
• Noncontractible periodic orbits in cotangent bundles and Floer homology
• On Non-contractible Periodic Orbits of Hamiltonian Diffeomorphisms
• Non-contractible Periodic Orbits in Hamiltonian Dynamics on Closed Symplectic Manifolds
• Non-contractible periodic orbits in Hamiltonian dynamics on tori
• An almost existence theorem for non-contractible periodic orbits in cotangent bundles
• Computation of annular capacity by Hamiltonian Floer theory of non-contractible periodic trajectories

Cited by in corpus (2)

• Minimax periodic orbits of convex Lagrangian systems on complete Riemannian manifolds
• Notes on symplectic squeezing in $T^{\ast }{\mathbb{T}}^n$ and spectra of Finsler dynamics