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Full measure reducibility for generic one-parameter family of quasi-periodic linear systems. (English) Zbl 1166.34019
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.

34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
34A30 Linear ordinary differential equations and systems
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