The absolute continuity of the conjugation of certain diffeomorphisms of the circle

0. Introduction Let /be an orientation preserving ^if'-diffeomorphism of the circle. If the rotation number a = p(/) is irrational and log Df is of bounded variation then, by a wellknown theorem of Denjoy,/is conjugate to the rigid rotation Ra. The conjugation means that there exists an essentially unique homeomorphism h of the circle such that / = h~Rah. The general problem of relating the smoothness of h to that of/ under suitable diophantine conditions on a has been studied extensively (cf. [Hi], [KO], [Y] and the references given there). At the bottom of the scale of smoothness for / there is a theorem of M. Herman [H2] which states that if Df is absolutely continuous and D log Df e V, p > 1, a = p(f) is of 'constant type' which means 'the coefficients in the continued fraction expansion of a are bounded', and if / is a perturbation of Ra, then h is absolutely continuous. Our purpose in this paper is to give a different proof and an improved version of Herman's theorem. The main difference in the result is that we do not need to assume that / is close to Ra; the proof is very different from Herman's and is very much in the spirit of [KO]. It is not hard to see that the condition of boundedness of the continued-fraction coefficients of a is essential. Given a with unbounded coefficients one can construct fe % such that h is purely singular (see e.g., [HS], [K], [L]). This paper assumes a general understanding of the dynamics of circle rotations. We shall refer to [KO] for some of the basic facts and notations (but not to the main results of [KO] which assume more smoothness of/ and give more for h).