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Fix a $\sigma$-finite atom-less measure $\mu$ on the unit circle, which is quasi-invariant and ergodic under the rotation $T$ of the angle $2\pi\theta$, $\theta$ irrational. By a well-known result of Krieger, we can have three alternatives:

(i) there is a probability measure $\nu\in[\mu]$ in the measure-class $[\mu]$ determined by $\mu$, which is invariant under the action of $T$;

(ii) there is a $\sigma$-finite, non finite measure $\nu\in[\mu]$ in the measure-class $[\mu]$ determined by $\mu$, which is invariant under the action of $T$;

(iii) none of the previous alternatives: namely, the measure-class $[\mu]$ contains no $\sigma$-finite invariant measure (always under the action of the irrational rotation $T$).

In the canonical crossed product construction, the three alternatives above exactly lead to von Neumann factors (because $\mu$ is ergodic) of type $II_1$, $II_\infty$ and $III$, respectively (and the type can be determined only by computing the $ratio\,set$ $r([\mu])$, depending only on the class $[\mu]$).

I'm searching for explicit examples corresponding to (iii) above relative specifically to the irrational rotation $T$.

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The question is answered in the paper:

S. Matsumoto: Orbit equivalence types of circle diffeomorphisms with a Liouville rotation number, Nonlinearity 26 (2013), 1401-1414,

at least for Liouville numbers.

The general case is covered by the Theorem in:

W. Krieger: On Borel Automorphismsand Their Quasi-Invariant Measures, Math. Z. 151 (1976), 19-24.

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