Full Measure Reducibility for Generic One-parameter Family of Quasi-periodic Linear Systems

Let C-omega(Lambda, gl(m, C)) be the set of m x m matrices A(lambda) depending analytically on a parameter. in a closed interval Lambda subset of R. Consider one- parameter families of quasi-periodic linear differential equations: (X) over dot = (A(lambda) + g(omega(1)t, ... , omega(r)t, lambda)) X, where A is an element of C-omega(Lambda, gl(m, C)), g is analytic and sufficiently small. We prove that there is an open and dense set A in C-omega(Lambda, gl(m, C)), such that for each A(lambda). A the equation can be reduced to an equation with constant coefficients by a quasi- periodic linear transformation for almost all lambda is an element of Lambda in Lebesgue measure sense provided that g is sufficiently small. The result gives an affirmative answer to a conjecture of Eliasson (In: Proceeding of Symposia in Pure Mathematics).