Action-Index Relations for Perfect Hamiltonian Diffeomorphisms

TL;DR

This paper investigates the relationships between actions and indices of fixed points in Hamiltonian diffeomorphisms, revealing algebraic conditions and extending results on periodic orbits in certain symplectic manifolds.

Contribution

It establishes new relations between actions and indices of fixed points under specific algebraic conditions and refines a previous result on the existence of large-period simple periodic orbits.

Findings

Actions and indices satisfy specific relations under algebraic conditions.

Refined the Conley conjecture for negative monotone symplectic manifolds.

Existence of arbitrarily large period simple periodic orbits for certain Hamiltonian diffeomorphisms.

Abstract

We show that the actions and indexes of fixed points of a Hamiltonian diffeomorphism with finitely many periodic points must satisfy certain relations, provided that the quantum cohomology of the ambient manifold meets an algebraic requirement satisfied for projective spaces, Grassmannians and many other manifolds. We also refine a previous result on the Conley conjecture for negative monotone symplectic manifolds, due to the second and third authors, and show that a Hamiltonian diffeomorphism of such a manifold must have simple periodic orbits of arbitrarily large period whenever its fixed points are isolated.