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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.

##### MSC:
 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
##### Keywords:
reducibility; quasi-periodic; KAM
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##### References:
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