Image of a shift map along the orbits of a flow

TL;DR

This paper investigates when a smooth orbit-preserving map between flows can be expressed as a smooth reparametrization of time, providing conditions under which such a reparametrization exists and characterizing flows with this property.

Contribution

The paper identifies a class of flows for which orbit-preserving maps can be smoothly reparametrized, extending understanding of flow reparametrizations and their relation to Lie group actions.

Findings

Flows with the specified property are characterized.

Any flow sharing the same orbits is a smooth reparametrization of the original.

The proof leverages results on diameters of effective Lie group actions.

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.