Abstract.
We prove that two C 3 critical circle maps with the same rotation number in a special set ? 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 set ? has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of C ∞ critical circle maps with the same rotation number that are not C 1+β conjugate for any β>0. The class of rotation numbers for which such examples exist contains Diophantine numbers.
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Received November 1, 1998 / final version received July 7, 1999
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de Faria, E., de Melo, W. Rigidity of critical circle mappings I. J. Eur. Math. Soc. 1, 339–392 (1999). https://doi.org/10.1007/s100970050011
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DOI: https://doi.org/10.1007/s100970050011