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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
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[1] Bogoljubov N.N., Mitropolski Ju.A., Samoilenko A.M.: Methods of Accelerated Convergence in Nonlinear Mechanics. Springer-Verlag, New York (1976)
[2] Coppel W.A.: Pseudo-autonomous linear equation. Bull. Austral. Math. Soc. 16, 61–65 (1977) · Zbl 0328.34007
[3] Dinaburg E.I., Sinai Y.G.: The one dimensional Schrödinger equation with a quasi-periodic potential. Funkt. Anal. i. Priloz 9, 8–21 (1975) · Zbl 0357.58011
[4] Eliasson L.H.: Floquet solutions for the one-dimensional quasi-periodic Schrödinger equation. Commun. Math. Phys. 146, 447–482 (1992) · Zbl 0753.34055
[5] Eliasson L.H.: Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum. Acta Math. 179, 153–196 (1997) · Zbl 0908.34072
[6] Eliasson, L.H.: Almost reducibility of linear quasi-periodic systems. In: Smooth Ergodic Theory and its Applications (Seattle, WA, 1999). Proc. Sympos. Pure Math., vol. 69, pp. 679–705. Amer. Math. Soc., Providence, RI (2001) · Zbl 1015.34028
[7] He H.L., You J.G.: An improved result for positive measure reducibility of quasi-periodic linear systems. Acta Math. Sin. (Engl. Ser.) 22(1), 77–86 (2006) · Zbl 1101.37016
[8] Johnson R.A., Moser J.: The rotation number for almost periodic potentials. Commun. Math. Phys. 84, 403–438 (1982) · Zbl 0497.35026
[9] Johnson R.A., Sell G.R.: Smoothness of spectral subbundles and reducibility of quasi-periodic linear differential systems. J. Diff. Eqn. 41, 262–288 (1981) · Zbl 0459.34021
[10] Jorba A., Simó C.: On the reducibility of linear differntial equations with quasi-periodic coefficients. J. Diff. Eq. 98, 111–124 (1992) · Zbl 0761.34026
[11] Kato T.: Perturbation Theory for Linear Operator. Springer-Verlag, New York Inc (1966) · Zbl 0148.12601
[12] Krikorian R.: Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts (French) [Reducibility of compact-group-valued skew-product systems]. Astérisque 259, 1–216 (1999)
[13] Krikorian R.: Réductibilité presque partout des flots fibrés quasi-périodiques á valeurs dans des groups compacts. Ann. Sci. École. Norm. Sup. 32, 187–240 (1999) · Zbl 1098.37510
[14] Krikorian R.: Global density of reducible quasi-periodic cocycles on \({\mathbb{T}^{1} \times SU(2)}\) . Ann. Math. 154, 269–326 (2001) · Zbl 1030.37003
[15] Lancaster P.: Theory of Matrices. Academic Press, NewYork and London (1969) · Zbl 0186.05301
[16] Moser J., Pöschel J.: An extension of a result by Dinaburg and Sinai on quasi-periodic potentials. Comment. Math. Helvetici. 59, 39–85 (1984) · Zbl 0533.34023
[17] Rüssmann H.: On the one dimensional Schrödinger equation with a quasi-periodic potential. Annal. NY Acad. Sci. 357, 90–107 (1980)
[18] Rychlik M.: Renormalization of cocycles and linaer ODE with almost-periodic coefficients. Invent. Math. 110(1), 173–206 (1992) · Zbl 0771.58013
[19] You J.: Perturbations of lower dimensional tori for Hamiltonian systems. J. Diff. Eq. 152, 1–29 (1999) · Zbl 0919.58055
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