Let \(C^{\omega}(\Lambda, gl(m, {\mathbb C}))\) be the set of \(m\times m\) matrices \(A(\lambda)\) depending analytically on a parameter \(\lambda\) in a closed interval \(\Lambda \subset {\mathbb R}\). The authors study the full measure reducibility of one-parameter families of quasi-periodic linear differential equations
\[ \dot{X} = (A(\lambda) + g(\omega_1 t,\dots, \omega_r t, \lambda)) X, \]
where \(A\in C^\omega(\Lambda, gl(m, {\mathbb C}))\), \(g\) is analytic and sufficiently small. The authors prove that there is an open and dense set \({\mathcal A}\) in \(C^{\omega}(\Lambda, gl(m, {\mathbb C}))\), such that for each \(A(\lambda)\in {\mathcal A}\) the equation can be reduced to an equation with constant coefficients by a quasi-periodic linear transformation for almost all \(\lambda \in \Lambda\) in Lebesgue measure sense provided that \(g\) 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 \(A(\lambda) + g(\varphi, \lambda)\), here \(\dot{\varphi} = \omega\), if \(A(\lambda)\) is of block diagonal form, one can find a linear transformation \(T(\varphi)\), which may not be close to the identity, to move some eigenvalues of \(A(\lambda)\) such that the resonance does not happen. Therefore, the transformed system \(\tilde{A}(\lambda) + \tilde{g}(\varphi, \lambda)\) satisfies the non-resonance conditions for all parameters, and the KAM type iterations can be done for all parameters.