First Betti number of weighted homology group of Hamiltonian vector fields on symplectic tori

TL;DR

This paper explores the homology groups of Hamiltonian vector fields on symplectic tori, incorporating weights to extend the theory beyond traditional symplectic vector spaces.

Contribution

It introduces a weighted homology framework for Hamiltonian vector fields on symplectic tori, expanding the understanding of their topological properties.

Findings

First Betti number computed for weighted homology groups

Demonstrates differences from classical homology in symplectic vector spaces

Provides new tools for studying Hamiltonian dynamics on tori

Abstract

Like (co)homology group theory of formal Hamiltonian vector fields on symplectic vector spaces, we try studying homology group theory on symplecit tori introducing the notion of weight.