Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems

In this paper, we prove that a quasi-periodic linear differential equation in sl(2,a"e) with two frequencies (alpha,1) is almost reducible provided that the coefficients are analytic and close to a constant. In the case that alpha is Diophantine we get the non-perturbative reducibility. We also obtain the reducibility and the rotations reducibility for an arbitrary irrational alpha under some assumption on the rotation number and give some applications for Schrodinger operators. Our proof is a generalized KAM type iteration adapted to all irrational alpha.