The Reducibility And Linearization Of Quasi-periodic Skew-product System

We mainly focus on the local and global rigidity of reducibility for quasiperiodi-cally linear skew product system(quasiperiodically linear Cocycle)and the lineariza-tion of quasiperiodically nonlinear system(quasiperiodically forced torus flow).In the first chapter,we will introduce basic notations and concepts that will be used in this thesis.First of all,we will present the related function spaces and norms.Then we will introduce our research object:quasiperiodically linear skew product system and quasiperiodically forced torus flow.Furthermore,we will introduce the basic concepts:Lyapunov exponent and rotation number,reducibility,linearization,and some concepts and results in number theory.Finally,we will present the analytic approximation of finitely differentiable functions and Z2 action.In the second chapter,we will discuss the local rigidity of reducibility of quasiperi-odically linear Cocycle.We recall some related results and methods of local rigidity of reducibility.Particularly,for Diophantine frequency we prove the local rigidity of reducibility of finitely differentiable U(n)-Cocycle.In the third chapter,we mainly consider the global rigidity of reducibility of quasiperiodically linear Cocycle.We will introduce the renormalization method in Cocycle and its applications.Based on the local rigidity of reducibility,applying the renormalization method,we will prove the global rigidity of reducibility of finitely differentiable U(n)-Cocycle with recurrence Diophantine frequency.In the fourth chapter,we will mainly discuss the linearization of quasiperiodically forced torus flow.For multi-parameter quasiperiodically forced torus flow(?,?(?)+F(?,?,?))(where ?=(1,?),? is irrational,??Rn is the parameter,?(?)is a vector-valued function with respect to A)we will show that under a suitable non-degenerate condition,there exists a positive measure set of ??Rn such that the system is C?rotations linearizable if F is small enough and a is not super-Liovillean.