A characterization of the standard Reeb flow

TL;DR

This paper characterizes the standard Reeb flow among continuous flows with planar Reeb foliation, showing it is uniquely determined by its conjugacy to all its positive time scalings.

Contribution

It provides a topological characterization of the standard Reeb flow based on conjugacy properties under time rescaling.

Findings

Standard Reeb flow is conjugate to all its positive time scalings.

Conjugacy to scaled flows characterizes the standard Reeb flow.

Provides a criterion for identifying the standard Reeb flow in topological terms.

Abstract

Among the topological conjugacy classes of the continuous flows $\{\phi^t\}$ whose orbit foliations are the planar Reeb foliation, there is one class called the standard Reeb flow. We show that $\{\phi^t\}$ is conjugate to the standard Reeb flow if and only if $\{\phi^t\}$ is conjugate to $\{\phi^{\lambda t}\}$ for any $\lambda>0$.