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

On the projective derivative cocycle for circle diffeomorphisms