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

Let f be an orientation preserving ℋ-diffeomorphism of the circle. If the rotation number α = ρ( f ) is irrational and log Df is of bounded variation then, by a wellknown theorem of Denjoy, f is conjugate to the rigid rotation R α . The conjugation means that there exists an essentially unique homeomorphism h of the circle such that f = h −l R α h . The general problem of relating the smoothness of h to that of f under suitable diophantine conditions on α has been studied extensively (cf. [H 1 ], [KO], [Y] and the references given there). At the bottom of the scale of smoothness for f there is a theorem of M. Herman [H 2 ] which states that if Df is absolutely continuous and D log Df ∈ L p , p > 1, α = ρ ( f ) is of ‘constant type’ which means ‘the coefficients in the continued fraction expansion of α are bounded’, and if f is a perturbation of R α , 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 f is close to R α ; the proof is very different from Herman's and is very much in the spirit of [KO].

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