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Let $v$ be continuous and nowhere-vanishing vector field tangent to the $n$-sphere $\mathbb{S}^n$ (hence $n$ is odd, w.r.t the Hairy-Ball Theorem). Let $x$ be a trajectory on $\mathbb{S}^n$, defined for $t\geq 0$, and solution of $\dot{x}=v(x)$.

Is this system ergodic or are there any other asymptotic possibilities (such that periodicity)?

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    $\begingroup$ 1) Ergodic with respect to which measure? 2) Beware of the Hopf fibration. $\endgroup$
    – D. Thomine
    Commented Oct 3, 2020 at 16:17
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    $\begingroup$ 1) Ergodic with respect to the Lebesgue measure. 2) I take a look a that, thank you! $\endgroup$
    – G. Panel
    Commented Oct 3, 2020 at 16:26
  • $\begingroup$ But there's no reason why specifically Lebesgue measure would be invariant. For example, you can move fast through some regions of $S^1$ and slowly through others. $\endgroup$
    – Christian Remling
    Commented Oct 3, 2020 at 18:06
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    $\begingroup$ $S^1$ acts freely and smoothly on every odd sphere $S^{2k-1}$ (regarded as the unit sphere in $\mathbb{C}^k$, with the action given by scalar multiplication) so generates a nowhere-vanishing vector field whose flow is very much non-ergodic. $\endgroup$
    – Qiaochu Yuan
    Commented Oct 3, 2020 at 18:41

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