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

Let C ω (Λ,gl(m,)) be the set of m×m matrices A(λ) depending analytically on a parameter λ in a closed interval Λ. The authors study the full measure reducibility of one-parameter families of quasi-periodic linear differential equations

X ˙=(A(λ)+g(ω 1 t,,ω r t,λ))X,

where AC ω (Λ,gl(m,)), g is analytic and sufficiently small. The authors prove that there is an open and dense set 𝒜 in C ω (Λ,gl(m,)), such that for each A(λ)𝒜 the equation can be reduced to an equation with constant coefficients by a quasi-periodic linear transformation for almost all λΛ 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(λ)+g(φ,λ), here φ ˙=ω, if A(λ) is of block diagonal form, one can find a linear transformation T(φ), which may not be close to the identity, to move some eigenvalues of A(λ) such that the resonance does not happen. Therefore, the transformed system A ˜(λ)+g ˜(φ,λ) satisfies the non-resonance conditions for all parameters, and the KAM type iterations can be done for all parameters.

MSC:
34C20Transformation and reduction of ODE and systems, normal forms
37J40Perturbations, normal forms, small divisors, KAM theory, Arnol’d diffusion
34C27Almost and pseudo-almost periodic solutions of ODE
34A30Linear ODE and systems, general
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