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From the paper

Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889.

the following result is known: Let $(E,\Sigma, \mu)$ be a measure space which is isomorphic to the unit interval. And let $G=G(E,\mu)$ be the space of all invertible measure preserving transformations on $(E,\Sigma, \mu)$. Then the subset of $G$ made up of weakly mixing maps is a dense $G_\delta$ set. In particular, from this it follows that the subset of $G$ made up of ergodic maps is a dense $G_\delta$ set.

Question: Can we similarly prove that ergodicity is generic in the set of measure-preserving $C^1$ diffeomorphisms of the circle, $\mathrm{Diff}^1(\mathbb T)$? If yes, what is a good reference for this?

Here by a map $T$ being ergodic I mean: for all $A\in\Sigma$, $$ T(A)=A \quad \Rightarrow \quad \mu(A)=0 \textrm{ or } \mu(A)=1; $$

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    $\begingroup$ The Halmos result that you quote requires that a measure is fixed in advance. If you fix Lebesgue measure, the diffeomorphisms preserving Lebesgue are the rigid rotations and rigid reflections. If the rotation is by an irrational angle, then of course Lebesgue is ergodic. Meanwhile, reflections (an open set) are never ergodic. If you didn’t mean to fix a measure, then I’m not sure what it means to say a diffeomorphism is ergodic. $\endgroup$
    – Anthony Quas
    Commented Sep 10, 2022 at 16:52

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