*Note: This is a concrete case of the following question: [Are almost all measure-preserving flows on compact manifolds ergodic?](https://mathoverflow.net/questions/408676/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?



Are $C^1$ vector fields generating an ergodic flow $C^0$ dense?
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.


