Rigidity of smooth critical circle maps

TL;DR

This paper proves that smooth critical circle maps with the same irrational rotation number of bounded type and odd criticality are conjugate via a smooth enough circle diffeomorphism, highlighting rigidity in their structure.

Contribution

It establishes a conjugacy result for critical circle maps with specific rotation numbers and criticality, advancing understanding of their rigidity properties.

Findings

Critical circle maps with same rotation number are conjugate.

Conjugacy is of class C^{1+α} for some universal α.

Results apply to maps with odd criticality and bounded type rotation numbers.

Abstract

We prove that any two $C^3$ critical circle maps with the same irrational rotation number of bounded type and the same odd criticality are conjugate to each other by a $C^{1+\alpha}$ circle diffeomorphism, for some universal $\alpha>0$.