Rigidity of critical circle maps

TL;DR

This paper proves that critical circle maps with the same irrational rotation number and criticality are conjugate by a $C^1$ diffeomorphism, with the conjugacy being $C^{1+eta}$ for almost all rotation numbers, highlighting rigidity properties.

Contribution

It establishes a rigidity result for critical circle maps, showing conjugacy under specific conditions and regularity of the conjugacy for almost all rotation numbers.

Findings

Conjugacy exists between maps with same rotation number and criticality.

Conjugacy is $C^{1+eta}$ for Lebesgue almost every rotation number.

Rigidity results extend understanding of critical circle map dynamics.

Abstract

We prove that any two $C^4$ critical circle maps with the same irrational rotation number and the same odd criticality are conjugate to each other by a $C^1$ circle diffeomorphism. The conjugacy is $C^{1+\alpha}$ for Lebesgue almost every rotation number.