Embedding of Analytic Quasi-Periodic Cocycles into Analytic Quasi-Periodic Linear Systems and its Applications

In this paper, we prove that any analytic quasi-periodic cocycle close to constant is the Poincar\'{e} map of an analytic quasi-periodic linear system close to constant. With this local embedding theorem, we get fruitful new results. We show that the almost reducibility of an analytic quasi-periodic linear system is equivalent to the almost reducibility of its corresponding Poincar\'e cocycle. By the local embedding theorem and the equivalence, we transfer the recent local almost reducibility results of quasi-periodic linear systems \cite{HoY} to quasi-periodic cocycles, and the global reducibility results of quasi-periodic cocycles \cite{A,AFK} to quasi-periodic linear systems. Finally, we give a positive answer to a question of \cite{AFK} and use it to prove Anderson localization results for long-range quasi-periodic operator with Liouvillean frequency, which gives a new proof of \cite{AJ05,AJ08,BJ02}. The method developed in our paper can also be used to prove some nonlinear local embedding results.Comment: 28 pages, no figur

Almost reducibility for finitely differentiable SL(2,R)-valued quasi-periodic cocycles

Quasi-periodic cocycles with a diophantine frequency and with values in SL(2,R) are shown to be almost reducible as long as they are close enough to a constant, in the topology of k times differentiable functions, with k great enough. Almost reducibility is obtained by analytic approximation after a loss of differentiability which only depends on the frequency and on the constant part. As in the analytic case, if their fibered rotation number is diophantine or rational with respect to the frequency, such cocycles are in fact reducible. This extends Eliasson's theorem on Schr\"odinger cocycles to the differentiable case

Resummation of perturbation series and reducibility for Bryuno skew-product flows

We consider skew-product systems on T^d x SL(2,R) for Bryuno base flows close to constant coefficients, depending on a parameter, in any dimension d, and we prove reducibility for a large measure set of values of the parameter. The proof is based on a resummation procedure of the formal power series for the conjugation, and uses techniques of renormalisation group in quantum field theory.Comment: 30 pages, 12 figure