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June  2016, 36(6): 3125-3152. doi: 10.3934/dcds.2016.36.3125

On local rigidity of reducibility of analytic quasi-periodic cocycles on $U(n)$

1. 

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

2. 

School of Science, Nanjing University of Science and Technology, Nanjing 210094, China

Received  June 2015 Revised  October 2015 Published  December 2015

We consider analytic cocycles on $\mathbb{T}^d\times U(n)$. We prove that, if a cocycle $(\alpha,A)$ with Diophantine $\alpha$ in an analytic class of radius $h$ can be conjugated to a constant cocycle $(\alpha,C)$ via some measurable conjugacy, then for almost all $C$, for any $h_*$ smaller than $h$, it can be conjugated to $(\alpha,C)$ in the analytic class of radius $h_*$, provided that $A$ is sufficiently close to some constant (the closeness depend only on $h-h_*$ and the Diophantine condition of $\alpha$).
Citation: Xuanji Hou, Lei Jiao. On local rigidity of reducibility of analytic quasi-periodic cocycles on $U(n)$. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3125-3152. doi: 10.3934/dcds.2016.36.3125
References:
[1]

A. Avila, Global theory of one-frequency Schrödinger operators I: Stratified analyticity of the Lyapunov exponent and the boundary of nonuniform hyperbolicity, Acta Mathematica, 215 (2015), 1-54. doi: 10.1007/s11511-015-0128-7.

[2]

A. Avila, Almost reducibility and absolute continuity I, arXiv:1006.0704, (2010).

[3]

A. Avila, B. Fayad and R. Krikorian, A KAM scheme for SL(2,R) cocycles with Liouvillean frequencies, Geom. Funct. Anal., 21 (2011), 1001-1019. doi: 10.1007/s00039-011-0135-6.

[4]

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

[5]

A. Avila and R. Krikorian, Reducibility or non-uniform hyperbolicity for quasiperiodic Schr\"odinger cocycles, Annals of Mathematics, 164 (2006), 911-940. doi: 10.4007/annals.2006.164.911.

[6]

N. Bogoljubov, J. Mitropolski and A. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer, New York, 1976.

[7]

C. Chavaudret, Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles, Bull. Soc. Math. France, 141 (2013), 47-106.

[8]

C. Chavaudret, Reducibility of quasi-periodic cocycles in Linear Lie groups, Ergod. Theory and Dyn. Syst., 31 (2010), 741-769.

[9]

E. Dinaburg and Y. Sinai, The one-dimensional Schrödinger equation with a quasi-periodic potential, Funct. Anal. Appl., 9 (1975), 279-289.

[10]

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

[11]

H. Eliasson, Almost reducibility of linear quasi-periodic systems, in Smooth Ergodic Theory and Its Applications (Seattle, WA., 1999), Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001, 679-705. doi: 10.1090/pspum/069/1858550.

[12]

B. Fayad and R. Krikorian, Rigidity results for quasiperiodic SL(2,R) cocycles, J. Mod. Dyn., 3 (2009), 479-510. doi: 10.3934/jmd.2009.3.479.

[13]

H. Her and J. You, Full measure reducibility for generic one-parameter family of quasi-periodic linear systems, Journal of Dynamics and Differential Equations, 20 (2008), 831-866. doi: 10.1007/s10884-008-9113-6.

[14]

X. Hou and L. Jiao, Full-measure uniformly analytic reducibility for one-parameter family of cocycles on $U(n)$, work in progress.

[15]

X. Hou and G. Popov, Rigidity of the reducibility of Gevrey quasi-periodic cocycles on U(n), To appear in Bulletin de la SMF, arXiv:1307.2954, (2013).

[16]

X. Hou and J. You, The local rigidity of reducibility of analytic quasi-periodic cocycles on $U(N)$, Discrete Contin. Dyn. Syst., 24 (2009), 441-454. doi: 10.3934/dcds.2009.24.441.

[17]

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

[18]

X. Hou and J. You, Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems, Inventiones Mathematicae, 190 (2012), 209-260. doi: 10.1007/s00222-012-0379-2.

[19]

R. Johnson and J. Moser, The rotation number for almost periodic potentials, J. Differ. Equ., 84 (1982), 403-438. doi: 10.1007/BF01208484.

[20]

A. Jorba and C. Simó, On the reducibility of linear differential equations with quasi-periodic coefficients, J. Differ. Equ., 98 (1992), 111-124. doi: 10.1016/0022-0396(92)90107-X.

[21]

N. Karaliolios, Differentiable Rigidity for quasiperiodic cocycles in compact Lie groups, arXiv:1407.4799, (2014), 269-326.

[22]

R. Krikorian, Réductibilité presque partout des flots fibrés quasi-périodiques à valeurs dans les groupes compacts, Ann. Sci. éc. Norm. Super., 32 (1999), 187-240. doi: 10.1016/S0012-9593(99)80014-7.

[23]

R. Krikorian, Réductibilité Des Systèmes Produits-croisés à Valeurs Das Des Groupes Compacts, Astérisque, 1999.

[24]

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

[25]

R. Krikorian, Reducibility, differentiable rigidity and Lyapunov exponents for quasi-periodic cocycles on $\mathbbT\times SL(2,\mathbbR)$, arXiv:math/0402333, (2004).

[26]

J. Moser and J. Pöschel, An extension of a result by Dinaburg and Sinai on quasiperiodic potentials, Comment. Math. Helv., 59 (1984), 39-85. doi: 10.1007/BF02566337.

[27]

H. Rüssmann, On the one dimensional schrödinger equation with a quasi-periodic potential, Ann. N.Y. Acad. Sci., 357 (1980), 90-107.

show all references

References:
[1]

A. Avila, Global theory of one-frequency Schrödinger operators I: Stratified analyticity of the Lyapunov exponent and the boundary of nonuniform hyperbolicity, Acta Mathematica, 215 (2015), 1-54. doi: 10.1007/s11511-015-0128-7.

[2]

A. Avila, Almost reducibility and absolute continuity I, arXiv:1006.0704, (2010).

[3]

A. Avila, B. Fayad and R. Krikorian, A KAM scheme for SL(2,R) cocycles with Liouvillean frequencies, Geom. Funct. Anal., 21 (2011), 1001-1019. doi: 10.1007/s00039-011-0135-6.

[4]

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

[5]

A. Avila and R. Krikorian, Reducibility or non-uniform hyperbolicity for quasiperiodic Schr\"odinger cocycles, Annals of Mathematics, 164 (2006), 911-940. doi: 10.4007/annals.2006.164.911.

[6]

N. Bogoljubov, J. Mitropolski and A. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer, New York, 1976.

[7]

C. Chavaudret, Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles, Bull. Soc. Math. France, 141 (2013), 47-106.

[8]

C. Chavaudret, Reducibility of quasi-periodic cocycles in Linear Lie groups, Ergod. Theory and Dyn. Syst., 31 (2010), 741-769.

[9]

E. Dinaburg and Y. Sinai, The one-dimensional Schrödinger equation with a quasi-periodic potential, Funct. Anal. Appl., 9 (1975), 279-289.

[10]

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

[11]

H. Eliasson, Almost reducibility of linear quasi-periodic systems, in Smooth Ergodic Theory and Its Applications (Seattle, WA., 1999), Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001, 679-705. doi: 10.1090/pspum/069/1858550.

[12]

B. Fayad and R. Krikorian, Rigidity results for quasiperiodic SL(2,R) cocycles, J. Mod. Dyn., 3 (2009), 479-510. doi: 10.3934/jmd.2009.3.479.

[13]

H. Her and J. You, Full measure reducibility for generic one-parameter family of quasi-periodic linear systems, Journal of Dynamics and Differential Equations, 20 (2008), 831-866. doi: 10.1007/s10884-008-9113-6.

[14]

X. Hou and L. Jiao, Full-measure uniformly analytic reducibility for one-parameter family of cocycles on $U(n)$, work in progress.

[15]

X. Hou and G. Popov, Rigidity of the reducibility of Gevrey quasi-periodic cocycles on U(n), To appear in Bulletin de la SMF, arXiv:1307.2954, (2013).

[16]

X. Hou and J. You, The local rigidity of reducibility of analytic quasi-periodic cocycles on $U(N)$, Discrete Contin. Dyn. Syst., 24 (2009), 441-454. doi: 10.3934/dcds.2009.24.441.

[17]

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

[18]

X. Hou and J. You, Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems, Inventiones Mathematicae, 190 (2012), 209-260. doi: 10.1007/s00222-012-0379-2.

[19]

R. Johnson and J. Moser, The rotation number for almost periodic potentials, J. Differ. Equ., 84 (1982), 403-438. doi: 10.1007/BF01208484.

[20]

A. Jorba and C. Simó, On the reducibility of linear differential equations with quasi-periodic coefficients, J. Differ. Equ., 98 (1992), 111-124. doi: 10.1016/0022-0396(92)90107-X.

[21]

N. Karaliolios, Differentiable Rigidity for quasiperiodic cocycles in compact Lie groups, arXiv:1407.4799, (2014), 269-326.

[22]

R. Krikorian, Réductibilité presque partout des flots fibrés quasi-périodiques à valeurs dans les groupes compacts, Ann. Sci. éc. Norm. Super., 32 (1999), 187-240. doi: 10.1016/S0012-9593(99)80014-7.

[23]

R. Krikorian, Réductibilité Des Systèmes Produits-croisés à Valeurs Das Des Groupes Compacts, Astérisque, 1999.

[24]

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

[25]

R. Krikorian, Reducibility, differentiable rigidity and Lyapunov exponents for quasi-periodic cocycles on $\mathbbT\times SL(2,\mathbbR)$, arXiv:math/0402333, (2004).

[26]

J. Moser and J. Pöschel, An extension of a result by Dinaburg and Sinai on quasiperiodic potentials, Comment. Math. Helv., 59 (1984), 39-85. doi: 10.1007/BF02566337.

[27]

H. Rüssmann, On the one dimensional schrödinger equation with a quasi-periodic potential, Ann. N.Y. Acad. Sci., 357 (1980), 90-107.

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