Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Zhicheng Tong, Shuyuan Xiao, Yong Li

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.

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

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.
Paper Structure (11 sections, 86 equations)

This paper contains 11 sections, 86 equations.

Table of Contents

  1. Introduction
  2. Modulus of continuity
  3. Cross-ratio estimates via modulus of continuity
  4. Circle diffeomorphisms
  5. Notations and lemmas
  6. Classical Denjoy's theory
  7. Denjoy-type inequality via modulus of continuity
  8. $C^1$ conjugacy through the convergence of $\sum\nolimits_{n = 0}^\infty {{k_{n + 1}}{\tau _n}}$
  9. $C^1$ conjugacy and further regularity through irrational rotation $\rho$ of $\varphi$-type
  10. Optimality of our integrability condition
  11. Appendix

Theorems & Definitions (11)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 1 more