Kuperberg Dreams

TL;DR

This paper explores the origins, properties, and open questions related to Kuperberg's smooth aperiodic flows on 3-manifolds, which serve as counterexamples to the Seifert Conjecture.

Contribution

It analyzes the genesis of Kuperberg's construction from Schweitzer's work and discusses known results and open problems in the dynamics of these flows.

Findings

Kuperberg flows are aperiodic and counterexamples to the Seifert Conjecture.

The paper reviews known dynamical properties of Kuperberg flows.

It highlights open questions about the ergodic and dynamical behavior of these flows.

Abstract

The ``Seifert Conjecture'' stated, ``Every non-singular vector field on the 3-sphere ${\mathbb S}^3$ has 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 discuss the genesi of Kuperberg's construction as an evolution from the Schweitzer construction of an aperiodic plug. We discuss some of the known results for Kuperberg flows, and discuss some of the many interesting questions and problems that remain open about their dynamical and ergodic properties.