Perspectives on Kuperberg flows. (English) Zbl 1454.37024

Hurder, Steven; Rechtman, Ana

Topol. Proc. 51, 197-244 (2018).

Summary: The “Seifert Conjecture” stated, “Every non-singular vector field on the 3-sphere \(\mathbb{S}^3\) has a periodic orbit”. In a celebrated work, [G. Kuperberg and K. Kuperberg, Ann. Math. (2) 143, No. 3, 547–576 (1996; Zbl 0856.57026)] 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.

MSC:

37C27	Periodic orbits of vector fields and flows
37C10	Dynamics induced by flows and semiflows
37C70	Attractors and repellers of smooth dynamical systems and their topological structure
37B40	Topological entropy
37B45	Continua theory in dynamics
37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior
57R30	Foliations in differential topology; geometric theory
58H05	Pseudogroups and differentiable groupoids

Keywords:

Kuperberg flow; aperiodic flow; minimal set; topological entropy; shape theory of minimal sets

Citations:

Zbl 0856.57026