Perspectives on Kuperberg flows

TL;DR

This paper reviews the development and properties of Kuperberg flows, highlighting how assumptions influence known results and discussing open questions about their dynamics and ergodic behavior.

Contribution

It provides a comprehensive overview of Kuperberg flows, analyzing how various assumptions affect existing results and identifying open problems in their dynamical and ergodic properties.

Findings

Kuperberg flows can be constructed to be aperiodic on 3-manifolds.

Known results depend heavily on specific assumptions about the flows.

Many open questions remain about the ergodic and dynamical properties of these flows.

Abstract

The "Seifert Conjecture" asks, "Does every non-singular vector field on the 3-sphere ${\mathbb S}^3$ have a periodic orbit?" In a celebrated work, Krystyna Kuperberg gave a construction of a smooth aperiodic vector field on a plug, which is then used to construct counter-examples to the Seifert Conjecture for smooth flows on the $3$-sphere, and on compact 3-manifolds in general. The dynamics of the flows in these plugs have been extensively studied, with more precise results known in special "generic" cases of the construction. Moreover, the dynamical properties of smooth perturbations of Kuperberg's construction have been considered. In this work, we recall some of the results obtained to date for the Kuperberg flows and their perturbations. Then the main point of this work is to focus attention on how the known results for Kuperberg flows depend on the assumptions imposed on the flows, and to discuss some of the many interesting questions and problems that remain open about their dynamical and ergodic properties.