Existence of a vector field with a finite number of limit cycles.

2010-06-18T20:07:42Z

The following question is related to the Seifert conjecture.

Let $M$ be a closed manifold with zero Euler characteristic. Is it true that each homotopy class of nowhere-zero vector fields on $M$ contains a vector field with a finite number of (stable) limit cycles (closed trajectories)? Is it easy to construct?

Answer by rpotrie for Existence of a vector field with a finite number of limit cycles.

2010-06-18T21:11:13Z

In any dimension bigger or equal to $4$, the answer is yes. See here.

In dimension 3, the question is adressed here. In fact, in this paper neccessary and sufficient conditions to be homotopic to a non singular Morse-Smale flow are given. Morse-Smale means to have finitely many non degenerate closed orbits and that those are al the non-wandering set. In dimension 3 there are restrictions to satisfy that property, however, this paper proves that one has in the homotopy class vector fields whose minimal sets consist of finitely many periodic orbits (maybe degenerate).

The reference is K. Yano, The homotopy class of non singular Morse Smale vector fields on 3 manifolds, Inventiones Math. (1985).

Existence of a vector field with a finite number of limit cycles. - MathOverflow

Existence of a vector field with a finite number of limit cycles.

Answer by rpotrie for Existence of a vector field with a finite number of limit cycles.