Conley conjecture and local Floer homology

TL;DR

This paper links algebraic properties of local Floer homology to Hamiltonian dynamics, proving the Conley conjecture for certain closed symplectic manifolds by showing the mean action spectrum is infinite.

Contribution

It introduces a novel connection between local Floer homology algebra and Hamiltonian dynamics, providing a new proof of the Conley conjecture for aspherical symplectic manifolds.

Findings

Product is non-uniformly nilpotent for isolated periodic orbits

Mean action spectrum of Hamiltonian diffeomorphisms with isolated orbits is infinite

Provides a simple proof of the Conley conjecture for closed aspherical symplectic manifolds

Abstract

In this paper we connect algebraic properties of the pair-of-pants product in local Floer homology and Hamiltonian dynamics. We show that for an isolated periodic orbit the product is non-uniformly nilpotent and use this fact to give a simple proof of the Conley conjecture for closed manifolds with aspherical symplectic form. More precisely, we prove that on a closed symplectic manifold the mean action spectrum of a Hamiltonian diffeomorphism with isolated periodic orbits is infinite.