Finite $C^{\infty }$-actions are described by a single vector field

Abstract

In this work it is shown that given a connected $C^{\infty }$-manifold $M$ of dimension $\ge 2$ and a finite subgroup $G\subset \mathrm{Diff}(M)$, there exists a complete vector field $X$ on $M$ such that its automorphism group equals $G\times \mathbb{R}$, where the factor $\mathbb{R}$ comes from the flow of $X$.