American Institute of Mathematical Sciences

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

Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition

Claire Chavaudret 1, and  Stefano Marmi 2,

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.
Keywords: reducibility, Quasiperiodic cocycle, KAM method., Brjuno condition, diophantine condition, rotation number.
Mathematics Subject Classification: Primary: 34C20; Secondary: 37CX.
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.  Google Scholar

[2]

A. D. Brjuno, An analytic form of differential equations, Math. Notes, 6 (1969), 927-931. doi: 10.1007/BF01146416.  Google Scholar

[3]

C. Chavaudret, Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles, to appear in Bull. Soc. Math. France, (2010), arXiv:0912.4814. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

A. Giorgilli and S. Marmi, Convergence radius in the Poincaré-Siegel problem, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 601-621.  Google Scholar

[7]

R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys., 84 (1982), 403-438. doi: 10.1007/BF01208484.  Google Scholar

[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.  Google Scholar

[9]

J. Pöschel, KAM à la R, Regul. Chaotic Dyn., 16 (2011), 17-23. doi: 10.1134/S1560354710520060.  Google Scholar

[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.  Google Scholar

[11]

J.-C. Yoccoz, "Petits Diviseurs en Dimension 1", Astérisque, 231, Société mathématique de France, 1995. Google Scholar

[12]

L.-S. Young, Lyapunov exponents for some quasi-periodic cocycles, Ergodic Theory Dynam. Systems, 17 (1997), 483-504. doi: 10.1017/S0143385797079170.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

A. D. Brjuno, An analytic form of differential equations, Math. Notes, 6 (1969), 927-931. doi: 10.1007/BF01146416.  Google Scholar

[3]

C. Chavaudret, Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles, to appear in Bull. Soc. Math. France, (2010), arXiv:0912.4814. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

A. Giorgilli and S. Marmi, Convergence radius in the Poincaré-Siegel problem, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 601-621.  Google Scholar

[7]

R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys., 84 (1982), 403-438. doi: 10.1007/BF01208484.  Google Scholar

[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.  Google Scholar

[9]

J. Pöschel, KAM à la R, Regul. Chaotic Dyn., 16 (2011), 17-23. doi: 10.1134/S1560354710520060.  Google Scholar

[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.  Google Scholar

[11]

J.-C. Yoccoz, "Petits Diviseurs en Dimension 1", Astérisque, 231, Société mathématique de France, 1995. Google Scholar

[12]

L.-S. Young, Lyapunov exponents for some quasi-periodic cocycles, Ergodic Theory Dynam. Systems, 17 (1997), 483-504. doi: 10.1017/S0143385797079170.  Google Scholar

[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.  Google Scholar

[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 & 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 & 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 & 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 & 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]

Suddhasattwa Das, Yoshitaka Saiki, Evelyn Sander, James A. Yorke. Solving the Babylonian problem of quasiperiodic rotation rates. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2279-2305. doi: 10.3934/dcdss.2019145
[8]

E. Muñoz Garcia, R. Pérez-Marco. Diophantine conditions in small divisors and transcendental number theory. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1401-1409. doi: 10.3934/dcds.2003.9.1401
[9]

Masaru Ikehata. On finding an obstacle with the Leontovich boundary condition via the time domain enclosure method. Inverse Problems & 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 & 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 & 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 & 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 & Management Optimization, 2021  doi: 10.3934/jimo.2021222
[14]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184
[15]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196
[16]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233
[17]

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 & Applied Analysis, 2017, 16 (2) : 699-718. doi: 10.3934/cpaa.2017034
[18]

Baojun Bian, Pengfei Guan. A structural condition for microscopic convexity principle. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 789-807. doi: 10.3934/dcds.2010.28.789
[19]

Shie Mannor, Vianney Perchet, Gilles Stoltz. A primal condition for approachability with partial monitoring. Journal of Dynamics & Games, 2014, 1 (3) : 447-469. doi: 10.3934/jdg.2014.1.447
[20]

Wenxian Shen. Global attractor and rotation number of a class of nonlinear noisy oscillators. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 597-611. doi: 10.3934/dcds.2007.18.597

2020 Impact Factor: 0.848

Tools

Metrics

  • PDF downloads (57)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy