Reducibility Of Quasi-Periodic Linear Skew-Products

In this thesis, we are devoted to studying the reducibility of quasi-periodic lin-ear skew-products, especially those arising from quasi-periodic Schrodinger operators. We are more concerned with almost reducibility of quasi-periodic linear skew-products with Liouvillean frequency, and it includes three important parts:local theory, global theory and semi-global theory.In the first chapter, we will give some basic notations and concepts of this the-sis. We will introduce our research object:quasiperiodic linear skew-product system, both the continuous version and the discrete version; then we will introduce the basic concepts:such as Lyapunov exponents and rotation number, reducibility and almost reducibility.In the second chapter, we will discuss the spectral theory of quasiperiodic Schrodinger operators; and the relationship with dynamical systems:the relationship between Lya-punov exponents, rotation number and the spectrum, reducibility and the spectrum; at last, we discuss the typical example:Almost Mathieu operator, we introduce Aubry duality, and discuss its spectral types and spectral structure.In the third chapter, we will mainly discuss the local embedding theorem, which means we prove that any analytic quasi-periodic cocycle close to constant is the Poincare map of an analytic quasi-periodic linear system close to constant. We also show that an analytic quasi-periodic linear system is almost reducible if and only if its corre-sponding Poincare cocycle is almost reducible. With these two results, we get fruitful new results, they will be introduced in the latter chapters. In the fourth chapter, we will mainly discuss the local reducibility theory of quasiperiodic linear skew-product system. We will introduce various reducibility results and methods both with Diophantine frequency and Liouvillean frequency. Especially, we will prove the local almost reducibility of GL(d, R) cocycle with Liouvillean fre-quency. As an application, we prove the Anderson localization results for long-range quasi-periodic operator with Liouvillean frequency.In the fifth chapter, we will mainly discuss the global reducibility theory of quasiperi-odic linear skew-product system. We will first introduce renormalization theory and its application, and give the global reducibility results both in the continuous case and discrete case; we then introduce Avila's global theory of one frequency quasiperiodic SL(2, R) cocycle, his Almost Reducibility Conjecture and recent research development.In the sixth chapter, we will mainly discuss the semi-global reducibility theory of quasiperiodic linear skew-product system. We will prove almost reducibility and non-perturbative reducibility for semi-global analytic quasi-periodic linear systems with two frequencies, which solved the Almost Reducibility Conjecture in the semi-global regime. As an application, we provide a counterexample to Kotani-Last's Conjecture.