Abstract
Abstract. We prove that two C3 critical circle maps with the same rotation number in a special set ± are C1+! conjugate for some !>0 provided their successive renormalizations converge together at an exponential rate in the C0 sense. The set ± has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of CX critical circle maps with the same rotation number that are not C1+# conjugate for any #>0. The class of rotation numbers for which such examples exist contains Diophantine numbers.