The Rotation Number for a Generic $C^1$ Family of Circle Diffeomorphisms is Not Hölder

We study the regularity of the rotation number as a function of the parameter in monotone one-parameter families of circle diffeomorphisms. We prove that for an open and dense set of $C^2$ monotone families, the optimal Hölder exponent is exactly $1\mathrm{/}2$, showing therefore that the earlier result by J. Graczyk is sharp. For a $C^1-$ dense set of $C^{1+\alpha }$ families, we show that the rotation number cannot be more regular than $\alpha \mathrm{/}(1+\alpha )-$Hölder. As a consequence, we get that the rotation number for a generic $C^1$ family of circle diffeomorphisms is not Hölder.