Herman's Theory Revisited (Extension)

TL;DR

This paper extends Herman's theory by proving that certain smooth circle diffeomorphisms with Diophantine rotation numbers are smoothly conjugate to rigid rotations, with explicit smoothness conditions.

Contribution

It establishes a new smoothness conjugacy result for circle diffeomorphisms with Diophantine rotation numbers, extending previous work on Herman's theory.

Findings

Proves conjugacy to rigid rotation under specified smoothness conditions.

Provides explicit smoothness bounds depending on Diophantine class and differentiability.

Enhances understanding of smooth dynamics on the circle.

Abstract

We prove that a $C^{3+\beta}$-smooth orientation-preserving circle diffeomorphism with rotation number in Diophantine class $D_\delta$, $0<\beta<\delta<1$, is $C^{2+\beta-\delta}$-smoothly conjugate to a rigid rotation.