Herman's theory revisited

Abstract

We prove that a C ^sup 2+α^-smooth orientation-preserving circle diffeomorphism with rotation number in Diophantine class D ^sub δ^, 0≤δ<α≤1, α-δ≠1, is C ^sup 1+α-δ^-smoothly conjugate to a rigid rotation. This is the first sharp result on the smoothness of the conjugacy. We also derive the most precise version of Denjoy's inequality for such diffeomorphisms.[PUBLICATION ABSTRACT]