Relation between irrationality and regularity for $ C^1 $ conjugacy of $ C^2 $ circle diffeomorphisms to rigid rotations
Dynamical Systems
2026-02-24 v2
Abstract
By introducing the modulus of continuity, we first establish the corresponding cross-ratio distortion estimates under 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 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.
Keywords
Cite
@article{arxiv.2211.01590,
title = {Relation between irrationality and regularity for $ C^1 $ conjugacy of $ C^2 $ circle diffeomorphisms to rigid rotations},
author = {Zhicheng Tong and Shuyuan Xiao and Yong Li},
journal= {arXiv preprint arXiv:2211.01590},
year = {2026}
}
Comments
27 pages. To appear in Chinese Ann. Math. Ser. B