The Calabi invariant and the least number of periodic solutions of locally Hamiltonian equations

TL;DR

This paper establishes a new lower bound on the number of one-periodic solutions for nondegenerate locally Hamiltonian equations on compact symplectic manifolds, linking it to Novikov homology and the Calabi invariant.

Contribution

It introduces a novel lower bound based on Betti numbers of Novikov homology, improving previous results and generalizing the homological Arnold conjecture.

Findings

Provides a lower bound related to Betti numbers of Novikov homology.

Improves upon previous bounds by Lê-Ono and Ono.

Generalizes the homological Arnold conjecture.

Abstract

In this paper we prove a lower bound for the least number of one-periodic solutions of nondegenerate locally Hamiltonian equations on compact symplectic manifolds in terms of the Betti numbers of the Novikov homology associated to the Calabi invariant of the locally Hamiltonian equations. Our result improves lower bounds obtained by L\^e-Ono and Ono for the least number of nondegenerate locally Hamiltonian symplectic fixed points. Our result also generalizes the homological Arnold conjecture proved by Fukaya-Ono and Liu-Tian.