# Are almost all measure-preserving flows on compact manifolds ergodic?

This may be a naive question, but I have been unable to find a reference that answers it directly, at least at a level that I can understand. My intuition from physics is that non-ergodicity is typically associated with conserved quantities, which should be atypical in generic systems without special symmetries, but I'm curious if there is a more rigorous way to arrive at this result (or a counterargument!)

• I'm not sure if this sounds plausible. If you do it the other way around (fix the dynamics, vary the measure), then the ergodic measures are exactly the extreme points of the invariant measures, so would be rare if you have more than one. Commented Nov 16, 2021 at 18:50
• What do you mean by "almost all" in this case? The space of flows is infinite-dimensional, hence, there is no canonical measure on it. Commented Nov 16, 2021 at 19:34
• For a slightly silly, but concrete example, consider a finite phase space, of cardinality $n$, with counting measure. Only a small fraction $1/n$ of the measure preserving transformations (in other words, all bijections) give you an ergodic transformation (the permutations consisting of a single cycle). Commented Nov 16, 2021 at 21:05
• Taking your comment as a definition, I think, the answer to your question depends on the details such as degree of differentiability and the precise topology you use. For instance, if you consider Hamiltonian flows on $S^2$, then KAM theory tells you that the density you are expecting fails, at least if you control enough derivatives, see here. On the opposite extreme, as a geometer, I like Lohkamp's theorem that Riemannian metrics with ergodic geodesic flows are dense in $C^0$-topology. Commented Nov 16, 2021 at 21:31
• I think Steve Alpern's work gives a positive answer to your question. Take a look at the book "Typical Dynamics of Volume Preserving Homeomorphisms" by Alpern and Prasad. Commented Nov 19, 2021 at 21:26

As it was mentioned before, KAM tells you that in the $$C^r$$-topology, for $$r$$ sufficiently large, ergodicity is not "typical".
For homeomorphism Oxtoby-Ulam proved here that ergodicity is $$C^0$$ typical.
For the $$C^1$$-topology there is a nice recent result by Avila-Crovisier-Wilkinson here which proves that for a $$C^1$$-typical volume preserving diffeomorphisms either you have zero (metric) entropy or you are ergodic. In this second case, they actually obtain something called non-uniformly Anosov (which gives more dynamical information).
In higher regularity ($$r>1$$) with more dynamical information something can be said. You can take a look at Pugh-Shub's conjecture for partially hyperbolic systems.