Herman's Theory Revisited

TL;DR

This paper proves that certain smooth circle diffeomorphisms with Diophantine rotation numbers are smoothly conjugate to rigid rotations and refines Denjoy's inequality for these maps.

Contribution

It establishes the smooth conjugacy under specific regularity and Diophantine conditions and provides the most precise version of Denjoy's inequality for such diffeomorphisms.

Findings

Proves $C^{1+eta}$ conjugacy for $C^{2+eta}$ diffeomorphisms with Diophantine rotation numbers.

Derives a sharp version of Denjoy's inequality for these diffeomorphisms.

Identifies the exact smoothness class of conjugacy based on regularity and Diophantine parameters.

Abstract

We prove that a $C^{2+\alpha}$-smooth orientation-preserving circle diffeomorphism with rotation number in Diophantine class $D_\delta$, $0<\delta<\alpha\le1$, is $C^{1+\alpha-\delta}$-smoothly conjugate to a rigid rotation. We also derive the most precise version of Denjoy's inequality for such diffeomorphisms.