American Institute of Mathematical Sciences

January  2012, 6(1): 59-78. doi: 10.3934/jmd.2012.6.59

Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition

 1 Département deMathématiques, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 02, France 2 Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126, Pisa (PI)

Received  November 2011 Published  May 2012

The arithmetics of the frequency and of the rotation number play a fundamental role in the study of reducibility of analytic quasiperiodic cocycles which are sufficiently close to a constant. In this paper we show how to generalize previous works by L.H. Eliasson which deal with the diophantine case so as to implement a Brjuno-Rüssmann arithmetical condition both on the frequency and on the rotation number. Our approach adapts the Pöschel-Rüssmann KAM method, which was previously used in the problem of linearization of vector fields, to the problem of reducing cocycles.
Citation: Claire Chavaudret, Stefano Marmi. Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition. Journal of Modern Dynamics, 2012, 6 (1) : 59-78. doi: 10.3934/jmd.2012.6.59
References:
 [1] 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. [2] A. D. Brjuno, An analytic form of differential equations, Math. Notes, 6 (1969), 927-931. doi: 10.1007/BF01146416. [3] C. Chavaudret, Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles, to appear in Bull. Soc. Math. France, (2010), arXiv:0912.4814. [4] L. H. Eliasson, Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation, Comm. Math. Phys., 146 (1992), 447-482. doi: 10.1007/BF02097013. [5] L. 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. [6] A. Giorgilli and S. Marmi, Convergence radius in the Poincaré-Siegel problem, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 601-621. [7] R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys., 84 (1982), 403-438. doi: 10.1007/BF01208484. [8] S. Marmi, P. Moussa and J.-C. Yoccoz, The Brjuno functions and their regularity properties, Comm. Math. Phys., 186 (1997), 265-293. doi: 10.1007/s002200050110. [9] J. Pöschel, KAM à la R, Regul. Chaotic Dyn., 16 (2011), 17-23. doi: 10.1134/S1560354710520060. [10] H. Rüssmann, KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 683-718. doi: 10.3934/dcdss.2010.3.683. [11] J.-C. Yoccoz, "Petits Diviseurs en Dimension 1", Astérisque, 231, Société mathématique de France, 1995. [12] L.-S. Young, Lyapunov exponents for some quasi-periodic cocycles, Ergodic Theory Dynam. Systems, 17 (1997), 483-504. doi: 10.1017/S0143385797079170. [13] J. Wang and Q. Zhou, Reducibility results for quasiperiodic cocycles with Liouvillean frequency, J. Dynam. Differential Equations, 24 (2011), 61-83. doi: 10.1007/s10884-011-9235-0.

show all references

References:
 [1] 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. [2] A. D. Brjuno, An analytic form of differential equations, Math. Notes, 6 (1969), 927-931. doi: 10.1007/BF01146416. [3] C. Chavaudret, Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles, to appear in Bull. Soc. Math. France, (2010), arXiv:0912.4814. [4] L. H. Eliasson, Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation, Comm. Math. Phys., 146 (1992), 447-482. doi: 10.1007/BF02097013. [5] L. 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. [6] A. Giorgilli and S. Marmi, Convergence radius in the Poincaré-Siegel problem, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 601-621. [7] R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys., 84 (1982), 403-438. doi: 10.1007/BF01208484. [8] S. Marmi, P. Moussa and J.-C. Yoccoz, The Brjuno functions and their regularity properties, Comm. Math. Phys., 186 (1997), 265-293. doi: 10.1007/s002200050110. [9] J. Pöschel, KAM à la R, Regul. Chaotic Dyn., 16 (2011), 17-23. doi: 10.1134/S1560354710520060. [10] H. Rüssmann, KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 683-718. doi: 10.3934/dcdss.2010.3.683. [11] J.-C. Yoccoz, "Petits Diviseurs en Dimension 1", Astérisque, 231, Société mathématique de France, 1995. [12] L.-S. Young, Lyapunov exponents for some quasi-periodic cocycles, Ergodic Theory Dynam. Systems, 17 (1997), 483-504. doi: 10.1017/S0143385797079170. [13] J. Wang and Q. Zhou, Reducibility results for quasiperiodic cocycles with Liouvillean frequency, J. Dynam. Differential Equations, 24 (2011), 61-83. doi: 10.1007/s10884-011-9235-0.
 [1] Claire Chavaudret, Stefano Marmi. Erratum: Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition. Journal of Modern Dynamics, 2015, 9: 285-287. doi: 10.3934/jmd.2015.9.285 [2] João Lopes Dias. Brjuno condition and renormalization for Poincaré flows. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 641-656. doi: 10.3934/dcds.2006.15.641 [3] Hans Koch, João Lopes Dias. Renormalization of diophantine skew flows, with applications to the reducibility problem. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 477-500. doi: 10.3934/dcds.2008.21.477 [4] Percy A. Deift, Thomas Trogdon, Govind Menon. On the condition number of the critically-scaled Laguerre Unitary Ensemble. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4287-4347. doi: 10.3934/dcds.2016.36.4287 [5] Saša Kocić. Reducibility of skew-product systems with multidimensional Brjuno base flows. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 261-283. doi: 10.3934/dcds.2011.29.261 [6] Michel Laurent, Arnaldo Nogueira. Rotation number of contracted rotations. Journal of Modern Dynamics, 2018, 12: 175-191. doi: 10.3934/jmd.2018007 [7] E. Muñoz Garcia, R. Pérez-Marco. Diophantine conditions in small divisors and transcendental number theory. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1401-1409. doi: 10.3934/dcds.2003.9.1401 [8] Suddhasattwa Das, Yoshitaka Saiki, Evelyn Sander, James A. Yorke. Solving the Babylonian problem of quasiperiodic rotation rates. Discrete and Continuous Dynamical Systems - S, 2019, 12 (8) : 2279-2305. doi: 10.3934/dcdss.2019145 [9] Masaru Ikehata. On finding an obstacle with the Leontovich boundary condition via the time domain enclosure method. Inverse Problems and Imaging, 2017, 11 (1) : 99-123. doi: 10.3934/ipi.2017006 [10] Lan Zeng, Guoxi Ni, Yingying Li. Low Mach number limit of strong solutions for 3-D full compressible MHD equations with Dirichlet boundary condition. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5503-5522. doi: 10.3934/dcdsb.2019068 [11] Hongxiu Zhong, Guoliang Chen, Xueping Guo. Semi-local convergence of the Newton-HSS method under the center Lipschitz condition. Numerical Algebra, Control and Optimization, 2019, 9 (1) : 85-99. doi: 10.3934/naco.2019007 [12] Abderrazak Chrifi, Mostafa Abounouh, Hassan Al Moatassime. Galerkin method of weakly damped cubic nonlinear Schrödinger with Dirac impurity, and artificial boundary condition in a half-line. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 79-93. doi: 10.3934/dcdss.2021030 [13] Jirui Ma, Jinyan Fan. On convergence properties of the modified trust region method under Hölderian error bound condition. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021222 [14] Xiayang Zhang, Yuqian Kong, Shanshan Liu, Yuan Shen. A relaxed parameter condition for the primal-dual hybrid gradient method for saddle-point problem. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022008 [15] Yuan Shen, Chang Liu, Yannian Zuo, Xingying Zhang. A modified self-adaptive dual ascent method with relaxed stepsize condition for linearly constrained quadratic convex optimization. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022101 [16] Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184 [17] Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196 [18] Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete and Continuous Dynamical Systems, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233 [19] TÔn Vı$\underset{.}{\overset{\hat{\ }}{\mathop{\text{E}}}}\,$T T$\mathop {\text{A}}\limits_.$, Linhthi hoai Nguyen, Atsushi Yagi. A sustainability condition for stochastic forest model. Communications on Pure and Applied Analysis, 2017, 16 (2) : 699-718. doi: 10.3934/cpaa.2017034 [20] Baojun Bian, Pengfei Guan. A structural condition for microscopic convexity principle. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 789-807. doi: 10.3934/dcds.2010.28.789

2021 Impact Factor: 0.641