Reducibility Of Quasi-periodic Cocycles On SL(2,R)

In the thesis,our main wok is devoted to studying the reducibility of quasi-periodic cocycles on SL(2,R),which is mainly to discuss and summarize its reducibil-ity.We mainly focus our attention on the rotations reducibility of quasi-periodic cocycles with Liouvillean frequency,and in our paper it contains two important sec-tions:the rotations reducibility of analytical cocycles and the rotations reducibility of Gevery cocycles.In the preface,we will introduce the history and the development process on the reducibility of quasi-periodic cocycles,and our main work in the thesis.In the second chapter,we are going to present some notations which we will use,and the related concepts of the paper.Firstly,we will give the research object in the thesis:the quasi-periodic cocycles on SL(2,R).After that,we will introduce the related concepts:conjugation,reducibility,fibred rotations number ect.In the end,we are going to give something about the knowledge of number theory and its properties.In the third chapter,we will present the norm,space of the norm and related estimates.Then we will give the important lemma and its corollary during the KAM iteration scheme.In the fourth chapter,we are going to debate on the reducibility of analytic cocycles on SL(2,R).In this part,we will prove that the analytic cocycle on SL(2,R)with Liouvillean frequency is locally rotations reducible.In the last chapter,our aim is to discuss the reducibility of Gevrey cocycles on SL(2,R).In this part,we will prove that the Gevery cocycle on SL(2,R)with Liouvillean frequency is locally rotations reducible.