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From

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.

Can a similar statement be proved if we replace in some way $G$ with the set of measure-preserving $C^1$ diffeomorphisms of the circle, $\mathrm{Diff}^1(\mathbb T)$?

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  • $\begingroup$ Are you asking if WM is generic inside $Diff^{1}$? With what topology? the $C^{1}$-topology? $\endgroup$
    – Asaf
    Commented Aug 29, 2022 at 18:25
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    $\begingroup$ Homeomorphisms of the circle are never weak mixing. $\endgroup$
    – Anthony Quas
    Commented Aug 29, 2022 at 19:10
  • $\begingroup$ @Asaf I don't know which topology is more appropriate to consider $\endgroup$
    – user490373
    Commented Aug 29, 2022 at 21:24
  • $\begingroup$ @AnthonyQuas Why not? And isn't this incompatible with Halmos' result? $\endgroup$
    – user490373
    Commented Aug 29, 2022 at 21:24
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    $\begingroup$ Given an orientation-preserving homeomorphism, it has a rotation number $a$. If $a$ is irrational, there is a factor map from $T$ to a rigid rotation by $a$. That precludes weak mixing. If $a$ is rational, any ergodic invariant measure is supported on a periodic orbit so is not weak-mixing. If $T$ is orientation-reversing, then $T^2$ is orientation-preserving, so not weak-mixing. $\endgroup$
    – Anthony Quas
    Commented Aug 30, 2022 at 5:40

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