Estimates of the linearization of circle diffeomorphisms

Abstract

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

Circle diffeomorphisms: quasi-reducibility and commuting diffeomorphisms

“…The following theorem gives an estimate of the linearization of a diffeomorphism having a rotation numbers of Diophantine constant type. This estimate, obtained in [2], is necessary to derive our results.…”

“…In order to derive our results, we use estimates of the conjugacy to rotations of diffeomorphisms having rotation numbers of Diophantine constant type. These estimates were obtained in [2].…”