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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 & Continuous Dynamical Systems - A, 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. doi: 10.1007/s11511-015-0128-7. Google Scholar [2] A. Avila, Almost reducibility and absolute continuity I,, , (2010). Google Scholar [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. doi: 10.1007/s00039-011-0135-6. Google Scholar [4] A. Avila and S. Jitomirskaya, Almost localization and almost reducibility,, J. Eur. Math. Soc., 12 (2010), 93. doi: 10.4171/JEMS/191. Google Scholar [5] A. Avila and R. Krikorian, Reducibility or non-uniform hyperbolicity for quasiperiodic Schr\"odinger cocycles,, Annals of Mathematics, 164 (2006), 911. doi: 10.4007/annals.2006.164.911. Google Scholar [6] N. Bogoljubov, J. Mitropolski and A. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics,, Springer, (1976). Google Scholar [7] C. Chavaudret, Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles,, Bull. Soc. Math. France, 141 (2013), 47. Google Scholar [8] C. Chavaudret, Reducibility of quasi-periodic cocycles in Linear Lie groups,, Ergod. Theory and Dyn. Syst., 31 (2010), 741. Google Scholar [9] E. Dinaburg and Y. Sinai, The one-dimensional Schrödinger equation with a quasi-periodic potential,, Funct. Anal. Appl., 9 (1975), 279. Google Scholar [10] H. Eliasson, Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation,, Comm. Math. Phys., 146 (1992), 447. doi: 10.1007/BF02097013. Google Scholar [11] H. Eliasson, Almost reducibility of linear quasi-periodic systems,, in Smooth Ergodic Theory and Its Applications (Seattle, (1999), 679. doi: 10.1090/pspum/069/1858550. Google Scholar [12] B. Fayad and R. Krikorian, Rigidity results for quasiperiodic SL(2,R) cocycles,, J. Mod. Dyn., 3 (2009), 479. doi: 10.3934/jmd.2009.3.479. Google Scholar [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. doi: 10.1007/s10884-008-9113-6. Google Scholar [14] X. Hou and L. Jiao, Full-measure uniformly analytic reducibility for one-parameter family of cocycles on $U(n)$,, work in progress., (). Google Scholar [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, (2013). Google Scholar [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. doi: 10.3934/dcds.2009.24.441. Google Scholar [17] X. Hou and J. You, The rigidity of reducibility of cocycles on $SO(N,\mathbbR)$,, Nonlinearity, 21 (2008), 2317. doi: 10.1088/0951-7715/21/10/006. Google Scholar [18] X. Hou and J. You, Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems,, Inventiones Mathematicae, 190 (2012), 209. doi: 10.1007/s00222-012-0379-2. Google Scholar [19] R. Johnson and J. Moser, The rotation number for almost periodic potentials,, J. Differ. Equ., 84 (1982), 403. doi: 10.1007/BF01208484. Google Scholar [20] A. Jorba and C. Simó, On the reducibility of linear differential equations with quasi-periodic coefficients,, J. Differ. Equ., 98 (1992), 111. doi: 10.1016/0022-0396(92)90107-X. Google Scholar [21] N. Karaliolios, Differentiable Rigidity for quasiperiodic cocycles in compact Lie groups,, , (2014), 269. Google Scholar [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. doi: 10.1016/S0012-9593(99)80014-7. Google Scholar [23] R. Krikorian, Réductibilité Des Systèmes Produits-croisés à Valeurs Das Des Groupes Compacts,, Astérisque, (1999). Google Scholar [24] R. Krikorian, Global density of reducible quasi-periodic cocycles on $\mathbbT^1\times SU(2)$,, Annals of Mathematics, 154 (2001), 269. doi: 10.2307/3062098. Google Scholar [25] R. Krikorian, Reducibility, differentiable rigidity and Lyapunov exponents for quasi-periodic cocycles on $\mathbbT\times SL(2,\mathbbR)$,, , (2004). Google Scholar [26] J. Moser and J. Pöschel, An extension of a result by Dinaburg and Sinai on quasiperiodic potentials,, Comment. Math. Helv., 59 (1984), 39. doi: 10.1007/BF02566337. Google Scholar [27] H. Rüssmann, On the one dimensional schrödinger equation with a quasi-periodic potential,, Ann. N.Y. Acad. Sci., 357 (1980), 90. Google Scholar

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##### 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. doi: 10.1007/s11511-015-0128-7. Google Scholar [2] A. Avila, Almost reducibility and absolute continuity I,, , (2010). Google Scholar [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. doi: 10.1007/s00039-011-0135-6. Google Scholar [4] A. Avila and S. Jitomirskaya, Almost localization and almost reducibility,, J. Eur. Math. Soc., 12 (2010), 93. doi: 10.4171/JEMS/191. Google Scholar [5] A. Avila and R. Krikorian, Reducibility or non-uniform hyperbolicity for quasiperiodic Schr\"odinger cocycles,, Annals of Mathematics, 164 (2006), 911. doi: 10.4007/annals.2006.164.911. Google Scholar [6] N. Bogoljubov, J. Mitropolski and A. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics,, Springer, (1976). Google Scholar [7] C. Chavaudret, Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles,, Bull. Soc. Math. France, 141 (2013), 47. Google Scholar [8] C. Chavaudret, Reducibility of quasi-periodic cocycles in Linear Lie groups,, Ergod. Theory and Dyn. Syst., 31 (2010), 741. Google Scholar [9] E. Dinaburg and Y. Sinai, The one-dimensional Schrödinger equation with a quasi-periodic potential,, Funct. Anal. Appl., 9 (1975), 279. Google Scholar [10] H. Eliasson, Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation,, Comm. Math. Phys., 146 (1992), 447. doi: 10.1007/BF02097013. Google Scholar [11] H. Eliasson, Almost reducibility of linear quasi-periodic systems,, in Smooth Ergodic Theory and Its Applications (Seattle, (1999), 679. doi: 10.1090/pspum/069/1858550. Google Scholar [12] B. Fayad and R. Krikorian, Rigidity results for quasiperiodic SL(2,R) cocycles,, J. Mod. Dyn., 3 (2009), 479. doi: 10.3934/jmd.2009.3.479. Google Scholar [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. doi: 10.1007/s10884-008-9113-6. Google Scholar [14] X. Hou and L. Jiao, Full-measure uniformly analytic reducibility for one-parameter family of cocycles on $U(n)$,, work in progress., (). Google Scholar [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, (2013). Google Scholar [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. doi: 10.3934/dcds.2009.24.441. Google Scholar [17] X. Hou and J. You, The rigidity of reducibility of cocycles on $SO(N,\mathbbR)$,, Nonlinearity, 21 (2008), 2317. doi: 10.1088/0951-7715/21/10/006. Google Scholar [18] X. Hou and J. You, Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems,, Inventiones Mathematicae, 190 (2012), 209. doi: 10.1007/s00222-012-0379-2. Google Scholar [19] R. Johnson and J. Moser, The rotation number for almost periodic potentials,, J. Differ. Equ., 84 (1982), 403. doi: 10.1007/BF01208484. Google Scholar [20] A. Jorba and C. Simó, On the reducibility of linear differential equations with quasi-periodic coefficients,, J. Differ. Equ., 98 (1992), 111. doi: 10.1016/0022-0396(92)90107-X. Google Scholar [21] N. Karaliolios, Differentiable Rigidity for quasiperiodic cocycles in compact Lie groups,, , (2014), 269. Google Scholar [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. doi: 10.1016/S0012-9593(99)80014-7. Google Scholar [23] R. Krikorian, Réductibilité Des Systèmes Produits-croisés à Valeurs Das Des Groupes Compacts,, Astérisque, (1999). Google Scholar [24] R. Krikorian, Global density of reducible quasi-periodic cocycles on $\mathbbT^1\times SU(2)$,, Annals of Mathematics, 154 (2001), 269. doi: 10.2307/3062098. Google Scholar [25] R. Krikorian, Reducibility, differentiable rigidity and Lyapunov exponents for quasi-periodic cocycles on $\mathbbT\times SL(2,\mathbbR)$,, , (2004). Google Scholar [26] J. Moser and J. Pöschel, An extension of a result by Dinaburg and Sinai on quasiperiodic potentials,, Comment. Math. Helv., 59 (1984), 39. doi: 10.1007/BF02566337. Google Scholar [27] H. Rüssmann, On the one dimensional schrödinger equation with a quasi-periodic potential,, Ann. N.Y. Acad. Sci., 357 (1980), 90. Google Scholar
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