Symplectic topology of Mañé's critical values

Abstract

We study the dynamics and symplectic topology of energy hypersurfaces of mechanical Hamiltonians on twisted cotangent bundles. We pay particular attention to periodic orbits, displaceability, stability and the contact type property, and the changes that occur at the Mane critical value c. Our main tool is Rabinowitz Floer homology. We show that it is defined for hypersurfaces that are either stable tame or virtually contact, and it is invariant under under homotopies in these classes. If the configuration space admits a metric of negative curvature, then Rabinowitz Floer homology does not vanish for energy levels k>c and, as a consequence, these level sets are not displaceable. We provide a large class of examples in which Rabinowitz Floer homology is non-zero for energy levels k>c but vanishes for k<c, so levels above and below c cannot be connected by a stable tame homotopy. Moreover, we show that for strictly 1/4-pinched negative curvature and non-exact magnetic fields all sufficiently high energy levels are non-stable, provided that the dimension of the base manifold is even and different from two.

101 pages, no figures, final version appeared in Geometry and Topology

References in corpus (3)

• Weak KAM theorem on non compact manifolds
• Symplectic aspects of Aubry-Mather theory
• Magnetic flows on Sol-manifolds: dynamical and symplectic aspects

Cited by in corpus (17)

• The contact geometry of the restricted 3-body problem
• Rabinowitz Floer homology: A survey
• Floer theory and reduced cohomology on open manifolds
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• The multiplicity problem for periodic orbits of magnetic flows on the 2-sphere
• Continuation homomorphism in Rabinowitz Floer homology for symplectic deformations
• On the existence of periodic orbits for magnetic systems on the two-sphere
• Trace formula for the magnetic Laplacian on a compact hyperbolic surface
• Invariance of symplectic cohomology and twisted cotangent bundles over surfaces
• Periodic orbits in virtually contact structures
• The Hopf-Rinow theorem and the Mañé critical value for magnetic geodesics on odd-dimensional spheres
• Displacement energy of unit disk cotangent bundles
• Periodic bounce orbits of prescribed energy
• Computing the Rabinowitz Floer homology of tentacular hyperboloids
• On a Theorem by Schlenk
• Bubbles and Onis
• Thermostats without conjugate points