Hyperbolic Fixed Points and Periodic Orbits of Hamiltonian Diffeomorphisms

TL;DR

This paper proves that certain closed monotone symplectic manifolds with hyperbolic fixed points of Hamiltonian diffeomorphisms necessarily have infinitely many periodic orbits, using Floer theory and quantum product properties.

Contribution

It establishes the existence of infinitely many periodic orbits for Hamiltonian diffeomorphisms with hyperbolic fixed points on a specific class of symplectic manifolds, extending previous results.

Findings

Hyperbolic fixed points imply infinitely many periodic orbits.

Energy bounds for Floer trajectories are independent of iteration.

Quantum product properties are crucial in the proof.

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.