• Previous Article
    The domain of analyticity of solutions to the three-dimensional Euler equations in a half space
  • DCDS Home
  • This Issue
  • Next Article
    $C^{\alpha}$-Hölder classical solutions for non-autonomous neutral differential equations
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, 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), 911-940. 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), 93-131. 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, New York, 1976.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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.  Google Scholar

[9]

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

[10]

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

[11]

G. Gallavotti and G. Gentile, Degenerate elliptic resonances, Comm. Math. Phys., 257 (2005), 319-362. 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), 317-357. 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, Ser. I, 343 (2006), 423-426.  Google Scholar

[14]

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.  Google Scholar

[15]

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.  Google Scholar

[16]

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

[17]

K. Khanin, J. Lopes Dias and J. Marklof, Multidimensional continued fractions, dynamic renormalization and KAM theory, Commun. Math. Phys., 270 (2007), 197-231. 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), 1559-1585. 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), 1131-1146. 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), 477-500.  Google Scholar

[21]

S. Kocić, Renormalization of Hamiltonians for Diophantine frequency vectors and KAM tori, Nonlinearity, 18 (2005), 1-32.  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), 187-240.  Google Scholar

[23]

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

[24]

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.  Google Scholar

[25]

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.  Google Scholar

[26]

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.  Google Scholar

[27]

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.  Google Scholar

[28]

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.  Google Scholar

[29]

J. Moser, Convergent series expansions for quasi-periodic motions, Mathematische Annalen, 169 (1967), 136-176. 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), 481-524. doi: doi:10.1017/S0143385705000362.  Google Scholar

[31]

M. Rychlik, Renormalization of cocycles and linear ODE with almost-periodic coefficients, Invent. Math., 110 (1992), 173-206. 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), 911-940. 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), 93-131. 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, New York, 1976.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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.  Google Scholar

[9]

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

[10]

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

[11]

G. Gallavotti and G. Gentile, Degenerate elliptic resonances, Comm. Math. Phys., 257 (2005), 319-362. 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), 317-357. 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, Ser. I, 343 (2006), 423-426.  Google Scholar

[14]

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.  Google Scholar

[15]

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.  Google Scholar

[16]

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

[17]

K. Khanin, J. Lopes Dias and J. Marklof, Multidimensional continued fractions, dynamic renormalization and KAM theory, Commun. Math. Phys., 270 (2007), 197-231. 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), 1559-1585. 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), 1131-1146. 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), 477-500.  Google Scholar

[21]

S. Kocić, Renormalization of Hamiltonians for Diophantine frequency vectors and KAM tori, Nonlinearity, 18 (2005), 1-32.  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), 187-240.  Google Scholar

[23]

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

[24]

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.  Google Scholar

[25]

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.  Google Scholar

[26]

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.  Google Scholar

[27]

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.  Google Scholar

[28]

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.  Google Scholar

[29]

J. Moser, Convergent series expansions for quasi-periodic motions, Mathematische Annalen, 169 (1967), 136-176. 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), 481-524. doi: doi:10.1017/S0143385705000362.  Google Scholar

[31]

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

[1]

Xuanji Hou, Lei Jiao. On local rigidity of reducibility of analytic quasi-periodic cocycles on $U(n)$. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3125-3152. doi: 10.3934/dcds.2016.36.3125

[2]

Xuanji Hou, Jiangong You. Local rigidity of reducibility of analytic quasi-periodic cocycles on $U(n)$. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 441-454. doi: 10.3934/dcds.2009.24.441

[3]

Jean Bourgain. On quasi-periodic lattice Schrödinger operators. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 75-88. doi: 10.3934/dcds.2004.10.75

[4]

Lei Jiao, Yiqian Wang. The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1585-1606. doi: 10.3934/cpaa.2009.8.1585

[5]

Meina Gao, Jianjun Liu. Quasi-periodic solutions for derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2101-2123. doi: 10.3934/dcds.2012.32.2101

[6]

Wenmeng Geng, Kai Tao. Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2977-2996. doi: 10.3934/dcdsb.2020216

[7]

Hans Koch, João Lopes Dias. Renormalization of diophantine skew flows, with applications to the reducibility problem. Discrete & Continuous Dynamical Systems, 2008, 21 (2) : 477-500. doi: 10.3934/dcds.2008.21.477

[8]

Jinhao Liang. Positive Lyapunov exponent for a class of quasi-periodic cocycles. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1361-1387. doi: 10.3934/dcds.2020080

[9]

Yingte Sun. Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021047

[10]

Àngel Jorba, Pau Rabassa, Joan Carles Tatjer. Superstable periodic orbits of 1d maps under quasi-periodic forcing and reducibility loss. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 589-597. doi: 10.3934/dcds.2014.34.589

[11]

P.E. Kloeden, Victor S. Kozyakin. The perturbation of attractors of skew-product flows with a shadowing driving system. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 883-893. doi: 10.3934/dcds.2001.7.883

[12]

Tomás Caraballo, Alexandre N. Carvalho, Henrique B. da Costa, José A. Langa. Equi-attraction and continuity of attractors for skew-product semiflows. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2949-2967. doi: 10.3934/dcdsb.2016081

[13]

Juan A. Calzada, Rafael Obaya, Ana M. Sanz. Continuous separation for monotone skew-product semiflows: From theoretical to numerical results. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 915-944. doi: 10.3934/dcdsb.2015.20.915

[14]

Sylvia Novo, Carmen Núñez, Rafael Obaya, Ana M. Sanz. Skew-product semiflows for non-autonomous partial functional differential equations with delay. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4291-4321. doi: 10.3934/dcds.2014.34.4291

[15]

Bogdan Sasu, A. L. Sasu. Input-output conditions for the asymptotic behavior of linear skew-product flows and applications. Communications on Pure & Applied Analysis, 2006, 5 (3) : 551-569. doi: 10.3934/cpaa.2006.5.551

[16]

Corrado Falcolini, Laura Tedeschini-Lalli. A numerical renormalization method for quasi–conservative periodic attractors. Journal of Computational Dynamics, 2020, 7 (2) : 461-468. doi: 10.3934/jcd.2020018

[17]

Claudia Valls. On the quasi-periodic solutions of generalized Kaup systems. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 467-482. doi: 10.3934/dcds.2015.35.467

[18]

Peng Huang, Xiong Li, Bin Liu. Invariant curves of smooth quasi-periodic mappings. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 131-154. doi: 10.3934/dcds.2018006

[19]

Qihuai Liu, Dingbian Qian, Zhiguo Wang. Quasi-periodic solutions of the Lotka-Volterra competition systems with quasi-periodic perturbations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1537-1550. doi: 10.3934/dcdsb.2012.17.1537

[20]

Yanling Shi, Junxiang Xu, Xindong Xu. Quasi-periodic solutions of generalized Boussinesq equation with quasi-periodic forcing. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2501-2519. doi: 10.3934/dcdsb.2017104

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (35)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]