# Irrational closed orbits of vector fields on $S^2$ (Limit cycles and trace formula)

Motivations: We first introduce our motivations: We wish to find an operator-theoretical interpretation for the number of limit cycles of a polynomial vector field on the plane. We quote the statement at page 9 of this paper which is the main motivation for our post

The formulas we want to consider in this section relate the compact orbits of a flow with the alternating sum of suitable traces on cohomology

Via Poincaré compactification, every planar polynomial vector field gives us a vector field $$X$$ on $$S^2$$ which has a lifting to a non vanishing $$S^1$$-invariant vector field $$\tilde{X}$$ on the total space $$S^3$$ of the Hopf fiber bundle $$S^3\to S^2$$ (see this question). The orthogonal complement of this vector field determines a $$2$$-dimensional distribution on $$S^3$$.

The Poincaré compactification is described at page 64 of the following paper [VG]. For a brief description of this method see the first paragraph of this post.

The purpose of this post is to apply the materials of the paper "Number theory and dynamical system in foliated spaces" to find a possible TRACE interpretation for the upper bound for the number of limit cycles of a planar polynomial vector field. In particular we wish to apply the materials of the first paragraph of page 11 of the above linked paper. So we lift a vector field $$X$$ on $$S^2$$ to a non-vanishing vector field $$\tilde{X}$$ on $$S^3$$, then we consider the $$2$$-dimensional distribution on $$S^3$$ which is perpendicular to $$\tilde{X}$$ and we wish to know if this distribution is an integrable distribution which is invariant under flow of $$\tilde{X}$$. But assuming that all things go well, which is not really the case, we have a problem as follows: "A relation between closed orbits of $$X$$ and closed orbits of $$\tilde{X}$$". It is possible that we lose our closed orbit of the base space after lifting to the total space. This is main motivation of this post. Now we describe our main question:

Main Question: We fix the standard Riemannian metric of $$S^3$$ and we consider the Hopf fibration $$P:S^3 \to S^2$$. Let $$Z$$ be the unit length tangent vector field to fibers of this fiber bundle. Assume that $$H\subset TS^3$$ be the horizontal space of this fiber bundle which is perpendicular $$Z$$. So for every $$x\in S^3$$, the linear map $$DP(x)$$ is a linear isomorphism between $$H_x$$ and $$T_{P(x) S^2}$$.

For a vector field $$X$$ on $$S^2$$ we consider the vector field $$\tilde{X}=Z+DP^{-1}(X)$$ on $$S^3$$. So $$DP(\tilde{X})=X$$ and $$\tilde{X}$$ is a non-vanishing invariant vector field on the total space $$S^3$$.

Is there an example of a vector field on $$S^2$$ which possesses a closed orbit $$\gamma$$, but the $$\tilde{X}$$-invariant torus $$P^{-1}(\gamma)$$ does not contain any closed orbit? (That is, the restriction of $$\tilde{X}$$ to $$P^{-1}(\gamma)$$ would define a Kronecker foliation, an irrational foliation of torus. (Any such closed orbit $$\gamma$$ is called an irrational closed orbit.

Is there an example of this situation with the extra assumption that $$X$$ is the Poincaré compactification of a polynomial vector field on $$\mathbb{R}^2$$?

[VG] Vidal, Claudio; Gómez, Pedro An extension of the Poincaré compactification and a geometric interpretation. Proyecciones 22 (2003), no. 3, 161–180.

• Have you already tried a vector field on $S^2$ with only two zeros and only one closed orbit, say, the equator? Because $\tilde X$ does not depend linearly on $X$, changing the length of $X$ should produce an example. – Sebastian Goette Jun 1 at 6:52
• @SebastianGoette Very interesting point. You mean that instead of $Z+\tilde{X}$ we consider $Z+f\tilde{X}$. Yes? To be honest I did not think to this situation of equator(Infinity). The reason is that it contains a singularity when we poincare compactify a planar algebraic vector field to a vector field on sphere.In this case every possible $Z+f\tilde{X}$ is not a Kronecker foliation. But your idea can apply to every regular closed orbit in the base space. So according to your comment i think that the answer to this post is Yes. – Ali Taghavi Jun 1 at 7:08
• @SebastianGoette So it would be interesting to consider the following version of conjectures and theorems in the linked paper: "For a 2 dimensional foliation of a 3 manifold which is invariant under the flow of a vector field $V$, can one introduce a trace formula which right hand side involves $V$- invariant torous( As well as non degenerate closed orbits). – Ali Taghavi Jun 1 at 7:12
• @SebastianGoette What about the following: We lift the vector field $X$ to $\tilde{X}$ but we dont add the vertical term $Z$. In this case can the preimage of a regular closed orbit be a Kronecker torus? – Ali Taghavi Jun 1 at 7:18
• To your last comment: now this is a question of the holonomy of the connection on the Hopf- (or tautological) bundle. If the periodic orbit is moved a little, the holonomy changes, and so it is possible to produce Kronecker foliations upstairs. – Sebastian Goette Jun 1 at 7:38