Image of a shift map along the orbits of a flow

Abstract

Let $(F_t)$ be a smooth flow on a smooth manifold $M$ and $h:M\to M$ be a smooth orbit preserving map. The following problem is studied: suppose that for every point $z$ of $M$ there exists a germ of a smooth function $f_z$ at $z$ such that near $z$ we have that $h(x)=F_{f_z(x)}(x)$. Can the functions $(f_z)$ be glued together to give a smooth function on all of $M$? This question is closely related to reparametrizations of flows. We describe a large class of flows for which the above problem can be resolved, and show that they have the following property: any smooth flow $(G_t)$ whose orbits coincides with the ones of $(F_t)$ is obtained from $(F_t)$ by smooth reparametrization of time. The proof of our principal statement uses results of D. Hoffman and L. N. Mann about diameters of effective actions of Lie grous of Riemannian manifolds.