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Zbl 1166.34019
Her, Hai-Long; You, Jiangong
Full measure reducibility for generic one-parameter family of quasi-periodic linear systems.
(English)
[J] J. Dyn. Differ. Equations 20, No. 4, 831-866 (2008). ISSN 1040-7294; ISSN 1572-9222/e

Let $C^{\omega}(\Lambda, gl(m, {\Bbb C}))$ be the set of $m\times m$ matrices $A(\lambda)$ depending analytically on a parameter $\lambda$ in a closed interval $\Lambda \subset {\Bbb R}$. 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 $A\in C^\omega(\Lambda, gl(m, {\Bbb C}))$, $g$ is analytic and sufficiently small. The authors prove that there is an open and dense set ${\cal A}$ in $C^{\omega}(\Lambda, gl(m, {\Bbb C}))$, such that for each $A(\lambda)\in {\cal A}$ 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 {\it 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.
[Jinzhi Lei (Beijing)]
MSC 2000:
*34C20 Transformation of ODE and systems
37J40 Perturbations, etc.
34C27 Almost periodic solutions of ODE
34A30 Linear ODE and systems

Keywords: reducibility; quasi-periodic; KAM

Citations: Zbl 1015.34028

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