Counting periodic orbits of vector fields over smooth closed manifolds

2020/12/03 by Eaman Eftekhary, Eftekhary, Eaman
Mathematics · #37C10 #Advanced Differential Equations and Dynamical Systems #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2012.01808

openalex publication_date 2020/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Γ of the moduli space of periodic and ghost orbits, we define an integer weight. When the vector field moves along a path, and Γ 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.

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