Small Divisors in One Dimension

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  • Format: Paperback
  • Copyright: 2014-04-30
  • Publisher: Cambridge Univ Pr
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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.

Table of Contents

Siegel's Theorem, Bruno Numbers and Quadratic Polynomials
Introduction to Part I.
Linearisation of univalent functions
Linearisation of quadratic polynomials
Centralisers and Differentiable Conjugation of Circle Diffeomorphisms
Introduction to Part II.
Notation and prerequisties
Linearised centraliser equation
Density of O infty_ alpha in F_ alpha^ infty
Continuous dependence of vector fields associated with germes of diffeomorphisms
On a class of diffeomorphisms of the interval
Centraliser of a diffeomorphism whose rotation number is rational
Centraliser of a diffeomorphism whose rotation number is irrationalfirst results
Centraliser of a diffeomorphism whose rotation number is irrational (continued)
Table of Contents provided by Publisher. All Rights Reserved.

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