Reducibility of linear equations with quasi-periodic coefficients. A survey

Preface This survey deals with some aspects of the problem of reducibility for linear equations with quasi-periodic coefficients. It is a compilation of results on this problem, some already classical and some other more recent. Our motivation comes from the study of stability of quasi-periodic motions and preservation of invariant tori in Hamiltonian mechanics (where the reducibility of linear equations with quasi-periodic coefficients plays an important role), and this has influenced much of the presentation. The first chapter is an introduction to linear equations with quasi-periodic coefficients and their reducibility. It settles notation that we will use along the survey and gives some motivations for the study of reducibility, as well as a discussion of Floquet theory for periodic systems and some results on the reducibility of linear scalar equations with quasi-periodic coefficients. The second chapter is a survey of results on exponential dichotomy and a related spectrum for linear skew-product flows. This theory applies to general quasi-periodic equations in any dimension, and some results and definitions given there will be used in the following chapters. A reducibility result is included. The third chapter is the longest of the whole survey and it deals with Schrödinger equation with quasi-periodic potential. It is divided into two parts. In the first one we study the classical spectral theory of self-adjoint operators both in the general case and in the case of a quasi-periodic potential. The ergodic invariants like the rotation number and the Lyapunov exponents are discussed in relation with the spectral properties of the Schrödinger operator. The second part is devoted to results on reducibility for this equation and some ideas about the proof, based on KAM techniques, are presented. Finally, some attention is paid to results on non-reducibility for this equation. The fourth chapter tries to give an overview of the existing results on the reducibility of linear equation with quasi-periodic coefficients in dimension greater than two. In the first part of the chapter we study results on reducibility of general quasi-periodic equations, and in the second one we focus on the case when the flow is defined on a compact group, more specifically SO(3, R), because almost all of these systems can be shown to be reducible close to constant coefficients. Finally we give some remarks on non-reducibility for these equations. The survey is not intended to be exhaustive and it is certainly not original, neither in the results …