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

## TL;DR

This paper extends Floer homology techniques to Finsler manifolds' cotangent bundles, establishing the existence of noncontractible periodic orbits under certain conditions and applying these results to various geometric and dynamical problems.

## Contribution

It generalizes BPS capacities to Finsler manifolds and proves new existence results for noncontractible periodic orbits in this setting.

## Key findings

- Existence of noncontractible periodic orbits for large Hamiltonians on Finsler cotangent bundles.
- Generalization of BPS capacities to Finsler geometry.
- Applications to geodesic length preservation, Hamiltonian dynamics, and symplectic squeezing.

## 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.