Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition

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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
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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. doi: 10.1007/s00039-011-0135-6. Google Scholar
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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. doi: 10.1007/s00039-011-0135-6. Google Scholar
[2]	A. D. Brjuno, An analytic form of differential equations,, Math. Notes, 6 (1969), 927. 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). Google Scholar
[4]	L. H. Eliasson, Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation,, Comm. Math. Phys., 146 (1992), 447. doi: 10.1007/BF02097013. Google Scholar
[5]	L. H. Eliasson, Almost reducibility of linear quasi-periodic systems,, in, 69 (2001), 679. Google Scholar
[6]	A. Giorgilli and S. Marmi, Convergence radius in the Poincaré-Siegel problem,, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 601. Google Scholar
[7]	R. Johnson and J. Moser, The rotation number for almost periodic potentials,, Comm. Math. Phys., 84 (1982), 403. 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. doi: 10.1007/s002200050110. Google Scholar
[9]	J. Pöschel, KAM à la R,, Regul. Chaotic Dyn., 16 (2011), 17. 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. doi: 10.3934/dcdss.2010.3.683. Google Scholar
[11]	J.-C. Yoccoz, "Petits Diviseurs en Dimension 1",, Astérisque, 231 (1995). Google Scholar
[12]	L.-S. Young, Lyapunov exponents for some quasi-periodic cocycles,, Ergodic Theory Dynam. Systems, 17 (1997), 483. 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. doi: 10.1007/s10884-011-9235-0. Google Scholar

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