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 $C^{1+α}$ conjugate for some $α>0$ provided their successive renormalizations converge together at an exponential rate in the $C^0$ sense. The number $α$ 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 $C^{1+β}$ conjugate for any $β>0$.