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

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?

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And you would like to allow periodic orbits that are not stable limit cycles? – Zarathustra Jun 18 2010 at 20:40
I would guess that the suspension of an Anosov diffeomorphism of the two torus is a counterexample, but I am not so sure now. – rpotrie Jun 18 2010 at 20:52
I would like that every closed trajectory is non-degenerate. But if it is more easier to construct a field without that restriction it would be also interesting. – Petya Jun 18 2010 at 20:52
In that case, it is the suspension of something isotopic to the identity, so it is homotopic to a suspension of a Morse-Smale diffeomorphisms in the torus. The suspension of the Anosov is not the 3-torus. You want finitely many closed orbits or finitely many which are stable?, I guess that in the latter case that should be true. – rpotrie Jun 18 2010 at 21:02
To get finitely many closed orbits, a finite number of "Wilson plugs"(Annals 1966) is enough : any orbit entering the plug is attracted to one of the two periodic orbits inside the plug, if it enters at a point in a certain "entry region" of the plug (with non-empty interior), and otherwise the orbit is "unchanged" : it re-emerges at the same point as before plugging. Now, by compactness, a finite number of annuli transverse to the flow suffice to meet all orbits. Thickening them a little and replacing by plugs gives you finitely many closed orbits. And the homotopy class is unchanged. – BS Jun 20 2010 at 14:59

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).

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