Hyperbolic Fixed Points and Periodic Orbits of Hamiltonian Diffeomorphisms

Abstract

We prove that for a certain class of closed monotone symplectic manifolds any Hamiltonian diffeomorphism with a hyperbolic fixed point must necessarily have infinitely many periodic orbits. Among the manifolds in this class are complex projective spaces, some Grassmannians, and also certain product manifolds such as the product of a projective space with a symplectically aspherical manifold of low dimension. A key to the proof of this theorem is the fact that the energy required for a Floer connecting trajectory to approach an iterated hyperbolic orbit and cross its fixed neighborhood is bounded away from zero by a constant independent of the order of iteration. This result, combined with certain properties of the quantum product specific to the above class of manifolds, implies the existence of infinitely many periodic orbits.

20 pages

References in corpus (4)

• On Non-contractible Periodic Orbits of Hamiltonian Diffeomorphisms
• A symplectic proof of a theorem of Franks
• Deformed Hamiltonian Floer theory, capacity estimates, and Calabi quasimorphisms
• Target-local Gromov compactness

Cited by in corpus (12)

• Hamiltonian Pseudo-rotations of Projective Spaces
• On Non-contractible Periodic Orbits of Hamiltonian Diffeomorphisms
• Non-contractible Periodic Orbits in Hamiltonian Dynamics on Closed Symplectic Manifolds
• Periodic Orbits of Hamiltonian Systems Linear and Hyperbolic at Infinity
• On hyperbolic points and periodic orbits of symplectomorphisms
• On Periodic Points of Symplectomorphisms on Surfaces
• On the generic behavior of the spectral norm
• On the Barcode Entropy of Reeb Flows
• Perfect Reeb flows and action-index relations
• Periodic points of rational area-preserving homeomorphisms
• Invariant Sets and Hyperbolic Closed Reeb Orbits
• On the Hofer-Zehnder conjecture for semipositive symplectic manifolds