The conjugating map for commutative groups of circle diffeomorphisms

For a single aperiodic, orientation preserving diffeomorphism on the circle, all known local results on the differentiability of the conjugating map are also known to be global results. We show that this does not hold for commutative groups of diffeomorphisms. Given a set of rotation numbers, we construct commuting diffeomorphisms in C^2-ε for all ε > 0 with these rotation numbers that are not conjugate to rotations. On the other hand, we prove that for a commutative subgroup ℱ ⊂ C^1+β, 0 < β < 1, containing diffeomorphisms that are perturbations of rotations, a conjugating map h exists as long as the rotation numbers of this subset jointly satisfy a Diophantine condition.