2
$\begingroup$

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)?

Improve this question
$\endgroup$
4
  • 3
    $\begingroup$ 1) Ergodic with respect to which measure? 2) Beware of the Hopf fibration. $\endgroup$
    – D. Thomine
    Oct 3, 2020 at 16:17
  • 1
    $\begingroup$ 1) Ergodic with respect to the Lebesgue measure. 2) I take a look a that, thank you! $\endgroup$
    – G. Panel
    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
    Oct 3, 2020 at 18:06
  • 3
    $\begingroup$ $S^1$ 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
    Oct 3, 2020 at 18:41

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.