NOTES ON A THEOREM OF KATZNELSON AND ORNSTEIN

Let log f ' be an absolutely continuous and f " =f ' is an element of L-p ( S-1, dl ) for some p > 1 ; where l is Lebesgue measure. We show that there exists a subset of irrational numbers of unbounded type, such that for any element (rho) over cap of this subset, the linear rotation R-(rho) over cap and the shift f(t) = f + t mod 1 ; t is an element of[0, 1) with rotation number (rho) over cap, are absolutely continuously conjugate. We also introduce a certain Zygmund-type condition depending on a parameter , and prove that in the case gamma > 1/2 there exists a subset of irrational numbers of unbounded type, such that every circle di eomorphism satisfying the corresponding Zygmund condition is absolutely continuously conjugate to the linear rotation provided its rotation number belongs to the above set. Moreover, in the case of gamma > 1 ; we show that the conjugacy is C-1-smooth.