Orbit equivalence types of circle diffeomorphisms with a Liouville rotation number

TL;DR

This paper demonstrates that for circle diffeomorphisms with a Liouville rotation number, all orbit equivalence types are densely represented within the space of smooth diffeomorphisms, highlighting the richness of their dynamical classifications.

Contribution

It proves the density of all orbit equivalence types in the space of smooth circle diffeomorphisms with a Liouville rotation number, extending understanding of their dynamical diversity.

Findings

All orbit equivalence types are $C^ abla$-dense for Liouville rotation numbers.

The result applies to subspaces of diffeomorphisms with different orbit types.

The work advances the classification of smooth circle diffeomorphisms based on orbit structure.

Abstract

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 $\alpha$.