Noncontractible periodic orbits in cotangent bundles and Floer homology

Abstract

For every nontrivial free homotopy class $\alpha$ of loops in every closed connected Riemannian manifold $M$, we prove existence of a noncontractible 1-periodic orbit, for every compactly supported time-dependent Hamiltonian on the open unit cotangent bundle which is sufficiently large over the zero section. The proof shows that the Biran-Polterovich-Salamon capacity is finite for every closed connected Riemannian manifold and every free homotopy class of loops. This implies a dense existence theorem for periodic orbits on level hypersurfaces and, consequently, a refined version of the Weinstein conjecture: Existence of closed characteristics (one associated to each nontrivial $\alpha$) on hypersurfaces in $T^{\ast }M$ which are of contact type and contain the zero section.

34 pages, 10 figures, submitted 15 April 2004

References in corpus (4)

• Functors and Computations in Floer homology with Applications Part II
• Floer homology and the heat flow
• Non-contractible periodic orbits, Gromov invariants, and Floer-theoretic torsions
• The Weinstein conjecture and the theorems of nearby and almost existence

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• Existence of noncontractible periodic orbits of Hamiltonian system separating two Lagrangian tori on $T^*\T^n$ with application to non convex Hamiltonian systems
• Topological Persistence in Geometry and Analysis
• Computation of annular capacity by Hamiltonian Floer theory of non-contractible periodic trajectories
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• Persistence modules, symplectic Banach-Mazur distance and Riemannian metrics
• Magnetic billiards and the Hofer-Zehnder capacity of disk tangent bundles of lens spaces
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• Non-contractible orbits for Hamiltonian functions on Riemann surfaces
• Barcode entropy for Reeb flows on contact manifolds with Liouville fillings
• On the Hofer-Zehnder capacity for twisted tangent bundles over closed surfaces