A Denjoy Theorem for commuting circle diffeomorphisms with mixed Hölder derivatives

Starting from the seminal works by Poincaré [13] and Denjoy [3], a deep theory for the dynamics of circle diffeomorphisms has been developed by many authors [1, 7, 8, 17], and most of the fundamental related problems have been already solved. Quite surprisingly, the case of several commuting diffeomorphisms is rater special, as it was pointed out for the first time by Moser [9] in relation to the problem of the smoothness for the simultaneous conjugacy to rotations. Roughly speaking, in this case it should be enough to assume a joint Diophantine condition on the rotation numbers which does not imply a Diophantine condition for any of them (see the recent work [5] for the solution of the C∞ case of Moser’s problem). A similar phenomenon concerns the classical Denjoy Theorem. Indeed, in [4] it was proved that if d ≥ 2 is an integer number and τ > 1/d, then the elements f1, . . . , fd of any family of C1+τ commuting circle diffeomorphisms are simultaneously (topologically) conjugate to rotations provided that their rotation numbers are independent over the rationals (that is, no non trivial linear combination of them with rational coefficients equals a rational number). In other words, the classical (and nearly optimal) C2 hypothesis for Denjoy Theorem can be weakened in the case of several commuting diffeomorphisms. The first and main result of this work is a generalization of this fact to the case of different regularities.