Small Divisors in One Dimension

by Yoccoz, J. C.

• ISBN13: 9780521662574
• ISBN10: 0521662575
• Format: Paperback
• Copyright: 2014-04-30
• Publisher: Cambridge Univ Pr

Summary

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)

Appendices

Bibliography

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