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Rigidity of critical circle mappings I

Rigidity of critical circle mappings I 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of the European Mathematical Society Springer Journals

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Publisher
Springer Journals
Copyright
Copyright © 1999 by Springer-Verlag Berlin Heidelberg & EMS
Subject
Mathematics; Mathematics, general
ISSN
1435-9855
DOI
10.1007/s100970050011
Publisher site
See Article on Publisher Site

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.

Journal

Journal of the European Mathematical SocietySpringer Journals

Published: Dec 1, 1999

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