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Does there exist a real analytic vector field on $S^3$ all of whose orbits are dense? The second paragraph of page 285 Of this paper says that there is a vector field whose almost all orbits are dense:

http://perso.ens-lyon.fr/ghys/articles/constructionchamps.pdf

Is there an example of such a vector field which belongs to the Lie algebra of all left-invariant vector fields of the Lie group $S^3$?

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    $\begingroup$ Every orbit of a left invariant vector field is closed (it is contained in the translate of a torus). $\endgroup$
    – abx
    Commented Jul 6, 2019 at 7:29
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    $\begingroup$ $SO(4)=SO(3) ⊕ SO(3)$, and pick two incommensurable circle actions. $\endgroup$
    – LeechLattice
    Commented Jul 6, 2019 at 7:54
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    $\begingroup$ Real-analytic counterexamples to Seifert conjecture were published by Kuperberg and Kuperberg in Annals of Math., 1996. $\endgroup$
    – Misha
    Commented Jul 6, 2019 at 9:42
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    $\begingroup$ @Ali Taghavi: the difference is that $S^3=SU(2)$ is semi-simple. Every element is contained in a circle. $\endgroup$
    – abx
    Commented Jul 6, 2019 at 10:28
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    $\begingroup$ OK, you are right. The vector field is actually on $SO(3)$ (= the unitary tangent bundle to $S^2$), hence it lifts to $S^3$. $\endgroup$
    – abx
    Commented Jul 6, 2019 at 14:21

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