Counting periodic orbits of vector fields over smooth closed manifolds

TL;DR

This paper introduces a method to count periodic orbits of vector fields on smooth closed manifolds by extending the space to include ghost orbits, and demonstrates the invariance of a defined weight under deformations.

Contribution

It defines a new integer weight for subsets of the moduli space of orbits, including ghost orbits, and proves its invariance under continuous deformations of the vector field.

Findings

The weight function remains constant when the vector field varies continuously.

Examples illustrate the application of the main theorem.

The approach extends the classical counting of periodic orbits to a broader setting.

Abstract

We address the problem of counting periodic orbits of vector fields on smooth closed manifolds. The space of non-constant periodic orbits is enlarged to a complete space by adding the ghost orbits, which are decorations of the zeros of vector fields. Associated with any compact and open subset $\Gamma$ of the moduli space of periodic and ghost orbits, we define an integer weight. When the vector field moves along a path, and $\Gamma$ deforms in a compact and open family, we show that the weight function stays constant. We also give a number of examples and computations, which illustrate the applications of our main theorem.