Reparametrizations of vector fields and their shift maps

Abstract

Let $M$ be a smooth manifold, $F$ be a smooth vector field on $M$, and $F_t$ be the local flow of $F$. Denote by $Sh(F)$ the space of smooth maps $h:M\to M$ of the following form: $h(x) = F_{f(x)}(x)$, where $f:M\to\mathbb{R}$ runs over all smooth functions on $M$ which can be substituted into the flow $F_t$ instead of time. This space often coincides with the identity component of the group of diffeomorphisms preserving orbits of $F$. In this note it is shown that $Sh(F)$ is not changed under reparametrizations and pushforwards of $F$. As an application it is proved that a vector field $F$ without non-closed orbits can be reparametrized to induce a circle action on $M$ if and only if there exists a smooth function $f:M\to (0,+\infty)$ such that for each non-singular point $x$ of $M$, the value $f(x)$ is an integer multiple of the period of $x$ with respect to $F$.