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Note: This is a concrete case of the following question: Are almost all measure-preserving flows on compact manifolds ergodic?

Let $M$ be a Riemannian manifold with its natural Riemannian measure, and $V$ a $C^1$ vector field on $M$ whose associated flow is measure preserving.

Does there exist, for every $\varepsilon > 0$, a $C^1$ vector field $W$ that is $\varepsilon$-close to $V$ in $C^0$ norm such that the flow generated by $W$ is measure preserving and ergodic?

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  • $\begingroup$ By the way, if the OP (of the linked question) was serious about "almost all", then asking for a dense set is a poor formalization, since then (for example) "almost all" real numbers are rational, or "almost all" continuous functions are polynomials etc. $\endgroup$
    – Christian Remling
    Commented Nov 17, 2021 at 16:20
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    $\begingroup$ The usual procedure on a complete metric space would be to ask for (at least) a dense $G_{\delta}$ set. $\endgroup$
    – Christian Remling
    Commented Nov 17, 2021 at 16:21
  • $\begingroup$ You probably want to add a reasonable assumption about the manifold so that it supports an ergodic flow, for instance, that your manifold is a torus of some dimension. The circle still provides a counter-example (because your $V$ can have stable zeroes). The first genuinely interesting case, I think, is $M=T^3$. $\endgroup$
    – Moishe Kohan
    Commented Nov 17, 2021 at 16:59

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You can't always approximate by ergodic flows, because ergodic flows might not even exist.

For example, on $S^2$ the Poincare-Bendixson theorem rules out ergodic flows, but there are many measure-preserving flows.

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