Let be the set of matrices depending analytically on a parameter in a closed interval . The authors study the full measure reducibility of one-parameter families of quasi-periodic linear differential equations
where , is analytic and sufficiently small. The authors prove that there is an open and dense set in , such that for each the equation can be reduced to an equation with constant coefficients by a quasi-periodic linear transformation for almost all in Lebesgue measure sense provided that is sufficiently small. The result gives an affirmative answer to a conjecture of L. H. Eliasson [Proc. Sympos. Pure Math. 69, 679–705 (2001; Zbl 1015.34028)].
The KAM method is applied to prove the result. However, the classical KAM method can only obtain a positive measure parameter set. To prove a full measure reducibility result, the authors improve the KAM iterative method so that at each KAM iteration step, one don’t need to discard any parameter whenever the non-resonant conditions are satisfied. For the original system , here , if is of block diagonal form, one can find a linear transformation , which may not be close to the identity, to move some eigenvalues of such that the resonance does not happen. Therefore, the transformed system satisfies the non-resonance conditions for all parameters, and the KAM type iterations can be done for all parameters.