A Survey In Diffeomorphisms Of The Circle

This survey outlines the rather well-studied theory concerning diffeomorphisms of the circle.The main goal of this survey is to present in detail the proofs of two profound theorems:Sinai-Khanin's C1 conjugation theorem and Arnold's theorem on analytic linearization.Not only the two theorems themselves are of theoretical interest,their proofs are of primary importance in the theory of dynamical system-s.The proof of Sinai-Khanin's theorem entails the main ideas of the method of renormalizations,and the proof of Arnold's theorem manifests the basic framework of KAM theory.The proofs given here adhere to the original papers,but are more careful than the original ones when dealing with details.In Section 1 we introduce the background and some basic notions of homeomor-phisms of the circle,including the rotation number and Poincare's lemma,etc.Also we state Sinai-Khanin's theorem and Arnold's theorem.Poincare's lemma serves as a basis for later discussions.We consider a prototype diffeomorphism of the circle in Section 2:irrational rotations,namely rotations of the circle by an irrational number.The study of irrational rotations yields naturally dynamical partitions of the circle.Solutions of many problems rely on meticulous estimates on such kind of partitions.As a consequence,we shall prove that irrational rotations are minimal.This fact will be useful in Section 3 when we prove Denjoy's theorem.Based on Section 2,we define for general diffeomorphisms of the circle their corresponding dynamical partitions,with the help of Poincare's lemma.From this we obtain Denjoy's lemma immediately,and hence we can prove the Denjoy's theorem.We prove Sinai-Khanin's theorem in Section 4.The key step is to define two sets of renormalization coordinates.The change of coordinates here is the most complicated part of the survey technically.But we wish to present the underlying simple and straightforward ideas via some pictures.In Section 5,we switch to another aspect of the theory of circle diffeomorphism:Arnold's theorem.We shall study analytic circle diffeomorphisms which are close to a rotation.And we impose some number theoretic conditions on the rotation number.These two points are common features in KAM theory.We hope that by proving the Arnold theorem,we demonstrate the basic ideas and a framework of proofs using KAM theory.