Kuperberg Dreams

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.