Estimates of the Linearization of Circle Diffeomorphisms

58 pagesA celebrated theorem by Herman and Yoccoz asserts that if the rotation number $\alpha$ of a $C^\infty$-diffeomorphism of the circle $f$ satisfies a Diophantine condition, then $f$ is $C^\infty$-conjugated to a rotation. In this paper, we establish explicit relationships between the $C^k$ norms of this conjugacy and the Diophantine condition on $\alpha$. To obtain these estimates, we follow a suitably modified version of Yoccoz's proof