## Institute for Mathematical Sciences

## Preprint ims97-16

** E. de Faria and W. de Melo**
* Rigidity of critical circle mappings I*

Abstract: We prove that two $C^r$ critical circle maps with the same rotation number of bounded type are $C^{1+\alpha}$ conjugate
for some $\alpha>0$ provided their successive renormalizations
converge together at an exponential rate in the $C^0$
sense. The number $\alpha$ 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
$C^{1+\beta}$ conjugate for any $\beta>0$.

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