Rigidity of critical circle mappings I

We prove that two $C^r$ critical circle maps with the same rotation numberof bounded type are $C^{1+\alpha}$ conjugate for some $\alpha>0$ providedtheir successive renormalizations converge together at an exponential rate inthe $C^0$ sense. The number $\alpha$ depends only on the rate of convergence.We also give examples of $C^\infty$ critical circle maps with the samerotation number that are not $C^{1+\beta}$ conjugate for any $\beta>0$.