Periodic orbit for certain vector field on $S^3$ (à la Seifert conjecture)

Question

The standard frame for $S^3$ consists of $X_i,X_j,X_k$ with $X_i(a)=ia, X_j(a)=ja, X_k(a)=ka$ where $i,j,k$ are standard quaternion numbers, $a\in S^3$, and the multiplication is the quaternion multiplication. This global frame on $S^3$ gives us a trivialization of $TS^3\simeq S^3\times \mathbb{R}^3$. So the Hopf map $P:S^3\to S^2\subset \mathbb{R}^3$ is counted as a unit vector field on $S^3$. This vector field is called "Hopf vector field".

Does the Hopf vector field have a periodic orbit? Is this vector field discussed in various attempts to find an analytic counter example to Seifert conjecture?

Answer 1

I think the question is perhaps confusing, since the term Hopf vector field is usually reserved for your $X_i$ vector field (which is tangent to the fibers of the standard Hopf fibration). As I understand, you are instead referring to the vector field which, at a point $p$ with $P(p) = (a,b,c) \in S^2 \subset \mathbb{R}^3$, is given by $Y := aX_i + bX_j + cX_k$. Assuming this is the correct interpretation, then the fiber over $(1,0,0)$ has $Y = X_i$ and so the fiber of $P$ over $(1,0,0)$ is a periodic orbit of $Y$.

Answer 2

Seifert actually proved that every vector field close enough to X_i has a periodic orbit. (In other words, if you perturbate slightly X_i whose all orbits are compact, then most orbits will of course in general become noncompact, but there will remain at least one compact orbit). K. Kuperberg built on S^3 a smooth (C^infty) nonsingular vector field without periodic orbit; W. Thurston noticed that one can even make Kuperberg's example real analytic. So, it is not "attempts" any more.