The conjugating map for commutative groups of circle diffeomorphisms (Q1912792)

According to Poincaré, to every orientation preserving homeomorphism \(f\) of the circle \(\mathbb{T}\) is associated a real parameter \(\alpha\in [0,1)\), called the rotation number, such that if \(f\) is aperiodic then the orbit structure of \(f\) is the same as the orbit structure of the rigid rotation \(R_\alpha\) of \(\mathbb{T}\) by \(\alpha\). Given a set of rotation numbers, the author constructs commuting diffeomorphisms of class \(C^{2-\varepsilon}\) with these rotation numbers that are not conjugate to rotations. On the other hand, he also proves that for a commutative subgroup of diffeomorphisms of \(\mathbb{T}\) of class \(C^{1+\beta}\), \(0<\beta<1\), containing diffeomorphisms that are perturbations of rotations, a conjugating map exists as long as the rotation numbers of this subset jointly satisfy a Diophantine condition.