Relation between irrationality and regularity for $ C^1 $ conjugacy of $ C^2 $ circle diffeomorphisms to rigid rotations

TL;DR

This paper investigates the relationship between irrationality and regularity in $ C^1 $ conjugacy of $ C^2 $ circle diffeomorphisms to rigid rotations, introducing new estimates and inequalities to understand their conjugacy properties.

Contribution

It introduces a novel approach using the modulus of continuity to establish cross-ratio distortion estimates and a near-optimal Denjoy-type inequality for $ C^2 $ smoothness, and links continuity with irrationality.

Findings

Established cross-ratio distortion estimates under $ C^2 $ smoothness.

Derived a near-optimal Denjoy-type inequality for circle diffeomorphisms.

Proved the regularity of conjugation is sharp and identified an explicit integrability correlation.

Abstract

By introducing the modulus of continuity, we first establish the corresponding cross-ratio distortion estimates under $ C^2 $ smoothness, and further derive a Denjoy-type inequality, which is almost optimal for dealing with circle diffeomorphisms. The latter plays a prominent role in the study of $ C^1 $ conjugacy to irrational rotations. We also establish an explicit integrability correlation between continuity and irrationality for the first time. Furthermore, the regularity of the conjugation is addressed and proved to be sharp.