On Conjugates and the Asymptotic Distortion of One-Dimensional C1+Bv Diffeomorphisms

We show that a C1+bv circle diffeomorphism with absolutely continuous derivative and irrational rotation number can be conjugated to diffeomorphisms that are C1+bv arbitrary close to the corresponding rotation. This improves a theorem of M. Herman, who established the same result but starting with a C2 diffeomorphism. We prove that the same holds for countable Abelian groups of circle diffeomorphisms acting freely, a result that is new even in the C∞ context. Related results and examples concerning the asymptotic distortion of diffeomorphisms are presented. Along this path, we provide a straightened version of the classical Denjoy-Kocsma inequality for absolutely continuous potentials. © The Author(s) 2021. Published by Oxford University Press. All rights reserved.