A Denjoy Type Theorem for Commuting Circle Diffeomorphisms with Derivatives Having Different Hölder Differentiability Classes

Let $d\ge2$ be an integer number, and let $f_k$, $k\in\{1,\dots,d\}$, be $C^{1+\tau_k}$ commuting circle diffeomorphisms, with $\tau_k\in]0,1[$ and $\tau_1+\cdots+\tau_d>1$. We prove that if the rotation numbers of the $f_k$'s are independent over the rationals (that is, if the corresponding action of $\mathbf Z^d$ on the circle is free), then they are simultaneously (topologically) conjugate to rotations.