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

Abstract

Abstract We show that a $C^{1+bv}$ circle diffeomorphism with absolutely continuous derivative and irrational rotation number can be conjugated to diffeomorphisms that are $C^{1+bv}$ arbitrary close to the corresponding rotation. This improves a theorem of M. Herman, who established the same result but starting with a $C^2$ diffeomorphism. We prove that the same holds for countable Abelian groups of circle diffeomorphisms acting freely, a result that is new even in the $C^{\infty }$ 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.$\hfill$In memory of Jean-Christophe Yoccoz