Advanced Search
Article Contents
Article Contents

Reducibility of skew-product systems with multidimensional Brjuno base flows

Abstract Related Papers Cited by
  • We develop a renormalization method that applies to the problem of the local reducibility of analytic skew-product flows on Td $\times$ SL(2,R). We apply the method to give a proof of a reducibility theorem for these flows with Brjuno base frequency vectors.
    Mathematics Subject Classification: Primary: 37E20; Secondary: 37C10, 37C60.


    \begin{equation} \\ \end{equation}
  • [1]

    A. Avila and R. Krikorian, Reducibility or nonuniform hyperbolity of quasiperiodic Schrödinger cocycles, Ann. Math., 164 (2006), 911-940.doi: doi:10.4007/annals.2006.164.911.


    A. Avila and S. Jitomirskaya, Almost localization and almost reducibility, Journal of the European Math. Soc., 12 (2010), 93-131.doi: doi:10.4171/JEMS/191.


    N. N. Bogoljubov, Ju. A. Mitropolitskii and A. M. Samoilenko, "Methods of Accelarated Convergence in Nonlinear Mechanics,'' Springer Verlag, New York, 1976.


    J. Bourgain, On the spectrum of lattice Schrödinger operators with deterministic potential, J. Anal. Math., 88 (2002), 221-254.doi: doi:10.1007/BF02786578.


    A. D. Brjuno, Analytic form of differential equations I, Trudy Moskov. Mat. Obshch. 25 (1971), 119-262.


    A. D. Brjuno, Analytic form of differential equations II, Trudy Moskov. Mat. Obshch. 26 (1972), 199-239.


    E. I. Dinaburg and Ja. G. Sinai, The one-dimensional Schrödinger equation with quasiperiodic potential, Funkcional. Anal. i Priložen., 9 (1975), 8-21.


    L. H. Eliasson, Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation, Comm. Math. Phys., 146 (1992), 447-482.doi: doi:10.1007/BF02097013.


    L. H. Eliasson, Linear quasi-periodic systems-reducibility and almost reducibility, in "XIVth International Congress on Mathematical Physics,'' World Sci. Publ., (2005), 195-205.


    G. Gallavotti, Twistless KAM tori, Comm. Math. Phys., 164 (1994), 145-156.doi: doi:10.1007/BF02108809.


    G. Gallavotti and G. Gentile, Degenerate elliptic resonances, Comm. Math. Phys., 257 (2005), 319-362.doi: doi:10.1007/s00220-005-1325-6.


    G. Gentile, Resummation of perturbation series and reducibility for Bryuno Skew-product flows, J. Stat. Phys., 125 (2006), 317-357.doi: doi:10.1007/s10955-006-9127-6.


    S. Hadj Amor, Sur la densité d'état de l'operateur de Schrödinger quasi-périodique unidimensionnel, C.R. Acad. Sci. Paris, Ser. I, 343 (2006), 423-426.


    E. Hille and R. S. Phillips, "Functional Analysis And Semi-Groups,'' AMS Colloquium Publications, 31, Rev. ed. of 1957, Amer. Math. Soc., Providence, RI, 1974.


    X. Hou and J. You, The rigidity of reducibility of cocycles on $\SO (N,\R)$, Nonlinearity, 21 (2008), 2317-2330.doi: doi:10.1088/0951-7715/21/10/006.


    S. B. Katok, Linear extensions of dynamical systems and the reducibility problem, Matematicheskie Zametki, 8 (1970), 451-462.


    K. Khanin, J. Lopes Dias and J. Marklof, Multidimensional continued fractions, dynamic renormalization and KAM theory, Commun. Math. Phys., 270 (2007), 197-231.doi: doi:10.1007/s00220-006-0125-y.


    H. Koch and S. Kocić, Renormalization of vector fields and Diophantine invariant tori, Ergod. Theor. Dynam. Sys., 28 (2008), 1559-1585.doi: doi:10.1017/S0143385707000892.


    H. Koch and S. Kocić, A renormalization group aproach to quasiperiodic motion with Brjuno frequencies, Ergod. Theor. Dynam. Sys., 30 (2010), 1131-1146.doi: doi:10.1017/S014338570900042X.


    H. Koch and J. Lopes Dias, Renormalization of Diophantine skew flows, with applications to the reducibility problem, Discrete Cont. Dyn. Sys., 21 (2008), 477-500.


    S. Kocić, Renormalization of Hamiltonians for Diophantine frequency vectors and KAM tori, Nonlinearity, 18 (2005), 1-32.


    R. Krikorian, Réducibilité presque partout des flots fibrés quasi-périodiques à valeurs dans des groupes compacts, Ann. Sci. de l'É.N.S. 4$^e$ série, 32 (1999), 187-240.


    R. Krikorian, Réducibilité des systèmes produits-croisés à valeurs dans des groupes compacts, Astérisque, 259 (1999), 1-216.


    R. Krikorian, $C^0$-densité globale des systèmes produits-croisés sur le cercle réductibles, Ergod. Theor. Dyn. Sys., 19 (1999), 61-100.doi: doi:10.1017/S0143385799120972.


    R. Krikorian, Global density of reducible quasi-periodic cocycles on $\T^1\times\SU(2)$, Ann. of Math., 154 (2001), 269-326.doi: doi:10.2307/3062098.


    J. C. Lagarias, Geodesic multidimensional continued fractions, Proc. London Math. Soc. (3), 69 (1994), 464-488.doi: doi:10.1112/plms/s3-69.3.464.


    J. Lopes Dias, A normal form theorem for Brjuno skew-systems through renormalization, J. Differential Equations, 230 (2006), 1-23.doi: doi:10.1016/j.jde.2006.07.021.


    J. Lopes Dias, Local conjugacy classes for analytic torus flows, J. Differential Equations, 245 (2008), 468-489.doi: doi:10.1016/j.jde.2008.04.006.


    J. Moser, Convergent series expansions for quasi-periodic motions, Mathematische Annalen, 169 (1967), 136-176.doi: doi:10.1007/BF01399536.


    J. Puig and C. Simó, Analytic families of reducible linear quasi-periodic differential equations, Ergod. Th. and Dynam. Sys., 26 (2006), 481-524.doi: doi:10.1017/S0143385705000362.


    M. Rychlik, Renormalization of cocycles and linear ODE with almost-periodic coefficients, Invent. Math., 110 (1992), 173-206.doi: doi:10.1007/BF01231330.

  • 加载中

Article Metrics

HTML views() PDF downloads(56) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint