×

Perspectives on Kuperberg flows. (English) Zbl 1454.37024

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

Citations:

Zbl 0856.57026
PDFBibTeX XMLCite
Full Text: arXiv