Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition

Previous Article
Hölder foliations, revisited
JMD Home
This Issue
Next Article
On primes and period growth for Hamiltonian diffeomorphisms

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

Abstract
Related Papers

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.
[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

Tools

Metrics

Other articles
by authors

on AIMS
Claire Chavaudret Stefano Marmi
on Google Scholar
Claire Chavaudret Stefano Marmi

[Back to Top]