Notes on a theorem of Katznelson and Ornstein
Abstract
Let $\log f'$ be an absolutely continuous and $f'/f'∈ L_{p}(S^{1}, d\ell)$ for some $p>1, $ where $\ell$ is Lebesgue measure. We show that there exists a subset of irrational numbers of unbounded type, such that for any element $\widehat{ρ}$ of this subset, the linear rotation $R_{\widehat{ρ}}$ and the shift $f_{t}=f+t\mod 1, $ $t∈ [0, 1)$ with rotation number $\widehat{ρ}, $ are absolutely continuously conjugate. We also introduce a certain Zygmund-type condition depending on a parameter $γ$, and prove that in the case $γ>\frac{1}{2}$ 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 $C^{1}$-smooth.