Dense properties of the space of the circle diffeomorphisms with a Liouville rotation number

TL;DR

This paper demonstrates that for any Liouville rotation number, certain subspaces of circle diffeomorphisms are densely populated in the $C^$ topology, highlighting the richness of the space of such diffeomorphisms.

Contribution

It establishes the $C^$-density of specific subspaces within the space of circle diffeomorphisms with a given Liouville rotation number, extending understanding of their structural properties.

Findings

Various subspaces are $C^$-dense in the space of circle diffeomorphisms with Liouville rotation number

The result applies to the space of orientation-preserving $C^$ diffeomorphisms

Highlights the abundance of certain dynamical behaviors in the space of circle diffeomorphisms

Abstract

Given any Liouville number $\alpha$, it is shown that various subspaces are $C^\infty$-dense in the space of the orientation preserving $C^\infty$ diffeomorphisms of the circle with rotation number $\alpha$.