JournalsjemsVol. 1, No. 4pp. 339–392

Rigidity of critical circle mappings I

  • Edson de Faria

    Universidade de São Paulo, Brazil
  • Welington de Melo

    Instituto de Matemática Pura e Aplicada, Rio De Janeiro, Brazil
Rigidity of critical circle mappings I cover
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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.

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Edson de Faria, Welington de Melo, Rigidity of critical circle mappings I. J. Eur. Math. Soc. 1 (1999), no. 4, pp. 339–392

DOI 10.1007/S100970050011