REDUCIBILITY OF SLOW QUASI-PERIODIC LINEAR SYSTEMS

In this note, we prove that the reducibility of analytic quasi-periodic linear systems close to constant is irrelevant to the size of the base frequencies. More precisely, we consider the quasi-periodic linear systems (X) over dot = (A + B(theta))X, (theta) over dot = lambda(-1)omega in C-m, where the matrix A is constant and omega is a fixed Diophantine vector, lambda is an element of R\{0}. We prove that the system is reducible for typical A if B(theta) is analytic and sufficiently small (depending on A, omega but not on lambda).