An example of Cooperberg - in the theory of dynamical systems - constructed by K. Cooperberg is a counterexample to the Seifert hypothesis . This is an example of an infinitely smooth vector field without singular points and periodic trajectories on a three-dimensional sphere. It is worth noting that all vector fields close enough to the Hopf bundle have periodic trajectories - this is stated by Seifert's theorem (which served as the motivation for the above hypothesis).
Design
An example of Kuperberg is constructed by rearranging a foliation with a finite number of periodic trajectories, which consists in gluing instead of a neighborhood of rectification a special vector field - a cork (or trap ) of Kuperberg . This last one is a vector field on a three-dimensional cube, vertical near the border and without special points inside, the Poincare map from the lower side of which to the upper side is identical everywhere where it is defined. Moreover, there are points on the lower face such that the trajectories entering the cube at these points never leave the cube.
When replacing a field in a rectification neighborhood around a section of a periodic trajectory with a Cooperberg trap, no new periodic trajectories are created (since the succession map has not changed globally), and the old periodic trajectory can be torn apart (it is enough to match the point of the old periodic trajectory to the point whose trajectory is “lost” inside the cube).
Generalizations
The Cooperberg construction also allows one to construct a smooth vector field without singular points and periodic trajectories on any closed three-dimensional manifold (as well as on closed manifolds of higher dimension, provided that the vector field without singular points exists at all - that the Euler characteristic of the manifold is equal to zero).
Links
- Étienne Ghys, Construction de champs de vecteurs sans orbite périodique (d'après Krystyna Kuperberg), Séminaire Bourbaki , Vol. 1993/94. Astérisque no. 227 (1995), Exp. No. 785, 5, 283-307.
- Krystyna Kuperberg. A smooth counterexample to the Seifert conjecture. Ann. of Math . (2) 140 (1994), no. 3, 723-732.
- K. Kuperberg, Aperiodic dynamical systems . Notices Amer. Math. Soc. 46 (1999), no. 9, 1035-1040.