Rigidity of critical circle mappings, I

Abstract

We prove that two $C^r$ critical circle maps with the same rotation number of bounded type are $C^{1+\alpha }$ conjugate for some $\alpha >0$ provided their successive renormalizations converge together at an exponential rate in the $C^0$ sense. The number $\alpha$ depends only on the rate of convergence. We also give examples of $C^{\infty }$ critical circle maps with the same rotation number that are not $C^{1+\beta }$ conjugate for any $\beta >0$.