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!)

## 1 Answer

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.

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