# 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?

• And you would like to allow periodic orbits that are not stable limit cycles? Commented 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. Commented 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. Commented 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. Commented 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.
Commented Jun 20, 2010 at 14:59

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