# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Bogoljubov N.N., Mitropolski Ju.A., Samoilenko A.M.: Methods of Accelerated Convergence in Nonlinear Mechanics. Springer-Verlag, New York (1976)  Coppel W.A.: Pseudo-autonomous linear equation. Bull. Austral. Math. Soc. 16, 61–65 (1977) · Zbl 0328.34007  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  Eliasson L.H.: Floquet solutions for the one-dimensional quasi-periodic Schrödinger equation. Commun. Math. Phys. 146, 447–482 (1992) · Zbl 0753.34055  Eliasson L.H.: Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum. Acta Math. 179, 153–196 (1997) · Zbl 0908.34072  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  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  Johnson R.A., Moser J.: The rotation number for almost periodic potentials. Commun. Math. Phys. 84, 403–438 (1982) · Zbl 0497.35026  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  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  Kato T.: Perturbation Theory for Linear Operator. Springer-Verlag, New York Inc (1966) · Zbl 0148.12601  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)  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  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  Lancaster P.: Theory of Matrices. Academic Press, NewYork and London (1969) · Zbl 0186.05301  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  Rüssmann H.: On the one dimensional Schrödinger equation with a quasi-periodic potential. Annal. NY Acad. Sci. 357, 90–107 (1980)  Rychlik M.: Renormalization of cocycles and linaer ODE with almost-periodic coefficients. Invent. Math. 110(1), 173–206 (1992) · Zbl 0771.58013  You J.: Perturbations of lower dimensional tori for Hamiltonian systems. J. Diff. Eq. 152, 1–29 (1999) · Zbl 0919.58055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.