This book falls into two parts. The first studies those circle
diffeomorphisms which commute with a diffeomorphism f whose
rotation number r(f) is irrational. When f is not smoothly
conjugated to a rotation, its rotation number r(f), according to
Herman's theory, cannot be Diophantine. The centralizer, f, thus
reflects how good are the rational and Diophantine approximations
of r. The second part is dedicated to the study of biholomorphisms
of a complex variable in the neighbourhood of a fixed point. In
1942 Siegel demonstrated that such a germ is analytically
linearizable if its linear part is a Diophantine rotation; the
arithmetic condition imposed on the rotation number was then
weakened by Bruno. A geometrical approach to the problem yields an
alternative proof of this result and also demonstrates the
converse. If the Bruno's arithmetic condition is not satisfied the
corresponding quadratic polynomial is not linearizable.
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