Orbit equivalence types of circle diffeomorphisms with a Liouville rotation number

This paper is concerned about the orbit equivalence types of $C^\infty$ diffeomorphisms of $S^1$ seen as nonsingular automorphisms of $(S^1,m)$, where $m$ is the Lebesgue measure. Given any Liouville number $\alpha$, it is shown that each of the subspace formed by type ${\rm II}_1$, ${\rm II}_\infty$, ${\rm III}_\lambda$ ($\lambda>1$), ${\rm III}_\infty$ and ${\rm III}_0$ diffeomorphisms are $C^\infty$-dense in the space of the orientation preserving $C^\infty$ diffeomorphisms with rotation number $\alpha$.