On the growth rate of periodic orbits for vector fields

TL;DR

This paper extends a classical result relating the growth rate of periodic orbits to topological entropy from surface diffeomorphisms to $C^1$ generic vector fields of any dimension, addressing challenges from singularities and flow shear.

Contribution

It generalizes Katok's result to $C^1$ vector fields in any dimension, overcoming difficulties posed by singularities and flow shear.

Findings

Established the relationship between periodic orbit growth and topological entropy for $C^1$ generic vector fields.

Extended classical results from surface diffeomorphisms to higher-dimensional vector fields.

Addressed technical challenges due to singularities and flow shear in the analysis.

Abstract

We establish the relationship between the growth rate of periodic orbits and the topological entropy for $C^1$ generic vector fields: this extends a classical result of Katok for $C^{1+\alpha}(\alpha>0)$ surface diffeomorphisms to $C^1$ generic vector fields of any dimension. The main difficulty comes from the existence of singularities and the shear of the flow.