Symplectic deformations of Floer homology and non-contractible periodic orbits in twisted disc bundles

Abstract

In this paper we establish the existence of periodic orbits belonging to any $\sigma$-atoroidal free homotopy class for Hamiltonian systems in the twisted disc bundle, provided that the compactly supported time-dependent Hamiltonian function is sufficiently large over the zero section and the magnitude of the weakly exact $2$-form $\sigma$ admitting a primitive with at most linear growth on the universal cover is sufficiently small. The proof relies on showing the invariance of Floer homology under symplectic deformations and on the computation of Floer homology for the cotangent bundle endowed with its canonical symplectic form. As a consequence, we also prove that, for any nontrivial atoroidal free homotopy class and any positive finite interval, if the magnitude of a magnetic field admitting a primitive with at most linear growth on the universal cover is sufficiently small, the twisted geodesic flow associated to the magnetic field has a periodic orbit on almost every energy level in the given interval whose projection to the underlying manifold represents the given free homotopy class. This application is carried out by showing the finiteness of the restricted Biran-Polterovich-Salamon capacity.

35 pages and 8 figures, final version, the referee's comments incorporated and references added

References in corpus (6)

• Functors and Computations in Floer homology with Applications Part II
• The Lusternik-Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles
• Non-contractible Periodic Orbits in Hamiltonian Dynamics on Closed Symplectic Manifolds
• Symplectic deformations of Floer homology and non-contractible periodic orbits in twisted disc bundles
• The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems
• Computation of annular capacity by Hamiltonian Floer theory of non-contractible periodic trajectories

Cited by in corpus (3)

• Symplectic deformations of Floer homology and non-contractible periodic orbits in twisted disc bundles
• Minimax periodic orbits of convex Lagrangian systems on complete Riemannian manifolds
• Infinitely many noncontractible closed magnetic geodesics on non-compact manifolds