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January  2011, 29(1): 261-283. doi: 10.3934/dcds.2011.29.261

Reducibility of skew-product systems with multidimensional Brjuno base flows

1. 

IMPA, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Brazil

Received  October 2009 Revised  February 2010 Published  September 2010

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.
Citation: Saša Kocić. Reducibility of skew-product systems with multidimensional Brjuno base flows. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 261-283. doi: 10.3934/dcds.2011.29.261
References:
[1]

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

[2]

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

[3]

N. N. Bogoljubov, Ju. A. Mitropolitskii and A. M. Samoilenko, "Methods of Accelarated Convergence in Nonlinear Mechanics,'', Springer Verlag, (1976).   Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

L. H. Eliasson, Linear quasi-periodic systems-reducibility and almost reducibility, in, XIVth International Congress on Mathematical Physics, (2005), 195.   Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

E. Hille and R. S. Phillips, "Functional Analysis And Semi-Groups,'', AMS Colloquium Publications, 31 (1957).   Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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.   Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

show all references

References:
[1]

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

[2]

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

[3]

N. N. Bogoljubov, Ju. A. Mitropolitskii and A. M. Samoilenko, "Methods of Accelarated Convergence in Nonlinear Mechanics,'', Springer Verlag, (1976).   Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

L. H. Eliasson, Linear quasi-periodic systems-reducibility and almost reducibility, in, XIVth International Congress on Mathematical Physics, (2005), 195.   Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

E. Hille and R. S. Phillips, "Functional Analysis And Semi-Groups,'', AMS Colloquium Publications, 31 (1957).   Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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.   Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

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