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Rigidity of critical circle mappings, I

Edson de Faria·Welington de Melo
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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 $β>0$.

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Edson de Faria, Welington de Melo. 1997-11-15. Rigidity of critical circle mappings, I. https://arxiv.org/abs/math/9711214

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