Notes on a theorem of Katznelson and Ornstein - Akhadkulov, Habibulla; Dzhalilov, Akhtam; Khanin, Konstantin (American Institute of Mathematical Sciences, 2017)
Let logf′ be an absolutely continuous and f′′/f′∈Lp(S1,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 ρˆ of this subset, the linear rotation Rρˆ and the shift ft=f+tmod1, t∈[0,1) with rotation number ρˆ, are absolutely continuously conjugate.We also introduce a certain Zygmund-type condition depending on a parameter γ, and prove that in the case γ>12 there exists a subset of irrational numbers of unbounded type, such that every circle diffeomorphism 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 γ>1, we show that the conjugacy is C1-smooth.