Reducibility For One-parameter Family Of Quasi-periodic Linear Systems

In this thesis,we will mainly discuss about the reducibility problems of one-parameter family of quasi-periodic linear systems from two aspects.On the one hand,under the conditions of Diophantine,we generate the positive measure reducibility and full measure reducibility of quasi-periodic linear systems;On the other hand,the Liouvillean frequency is considered,we prove the positive measure diagonal reducibil-ity of a general high dimensional quasi-periodic linear system,it is a generalization of the rotation reducibility.The concrete content is in the following:In the first chapter,we introduce the research background of quasi-periodic linear systems and the problems to be studied in this paper.In the second chapter,we introduce some preliminary knowledge.The first section introduces some basic concepts involved in this thesis.The second section introduces the definition of the relevant norms for this paper and gives the relationship between the norms.In the third section,we give an important lemma and its proof.The lemma is very important to the proof of the later several principal theorems.The last section introduces the related knowledge of continued fractions.In the third chapter,we study the positive measure reducibility for one-parameter family of quasi-periodic linear systems under Diophantine frequency,the proof method here is different from the existing proof methods.In the fourth chapter,under the Diophantine frequency,we study the full mea-sure analtic reducibility for analytic one-parameter family of quasi-periodic linear systems on the basis of the positive measure reducibility of the third chapter.The proof is mainly based on the important lemma of the second chapter and lebesgue density theorem to simplify the iteration and improve the existing results,the ex-isting result is full measure C? reducibility,and this paper obtains the full measure analytic reducibility and the analytic radius of conjugate can be arbitrarily close to the analytic radius of the original system about ?.In the fifth chapter,we study the positive measure diagonal reducibility of a high-dimensional quasi-periodic linear system under the Liouvillean frequency.The proof method is mainly based on rotation reducibility in Hou-You[8].