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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

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

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