On the projective derivative cocycle for circle diffeomorphisms

TL;DR

This paper investigates the projective derivative cocycle for circle diffeomorphisms, providing explicit formulas and exploring conditions for reducibility to rotations, with extensions to higher-dimensional actions.

Contribution

It introduces explicit expressions for the projective derivative cocycle and extends the analysis to the 3-torus, advancing understanding of reducibility in this context.

Findings

Derived explicit formulas for the cocycle

Established conditions for reducibility to rotations

Extended results to the 3-torus setting

Abstract

We study the projective derivative as a cocycle of M\"obius transformations over groups of circle diffeomorphisms. By computing precise expressions for this cocycle, we obtain several results about reducibility and almost reducibility to a cocycle of rotations. We also introduce an extension of this cocycle to the diagonal action on the 3-torus for which we generalize the previous results.