Commuting circle diffeomorphisms with their derivatives having mixed moduli of continuity

Abstract

Let $d\ge 2$ be an integer and let ${\omega }_1,\cdots ,{\omega }_d$ be moduli of continuity in a specified class which contains the moduli of Hölder continuity. Let $f_k$, $k\in \{1,\cdots ,d\}$, be $C^{1+{\omega }_k}$ orientation preserving diffeomorphisms of the circle and $f_1,\cdots ,f_d$ commute with each other. We prove that if the rotation numbers of $f_k$'s are independent over the rationals and ${\omega }_1(t)\cdots {\omega }_d(t)=t\omega (t)$ with ${\lim }_{t\to 0^{+}}\omega (t)=0$, then $f_1,\cdots ,f_d$ are simultaneously (topologically) conjugate to rigid rotations.