Ergodicity of a dynamical system on the $n$-sphere

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)?

• 1) Ergodic with respect to which measure? 2) Beware of the Hopf fibration. Oct 3, 2020 at 16:17
• 1) Ergodic with respect to the Lebesgue measure. 2) I take a look a that, thank you! Oct 3, 2020 at 16:26
• 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. Oct 3, 2020 at 18:06
• $S^1$ 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. Oct 3, 2020 at 18:41