Absolute Continuity of the Rotation Number for Quasi-Periodic CO-Cycles in \(SL(2, \mathbb {

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• 作者：Sana Hadj Amor (1)
  
• 关键词：Rotation number ; K.A.M. theory ; Absolutely continuous function ; 37E45 ; 35B20 ; 34A30
• 刊名：Mathematical Physics, Analysis and Geometry
• 出版年：2014
• 出版时间：June 2014
• 年：2014
• 卷：17
• 期：1-2
• 页码：151-167
• 全文大小：461 KB
• 参考文献：1. Avila, A., Damanik, D.: Absolute continuity of the integrated density of states for the almost Mathieu operator with non-critical coupling.
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  7. Amor, Hadj: Sana: H?lder continuity of the rotation number for quasi-periodic co-cycles in \(SL(2, \mathbb R)\) . Commun. Math. Phys. 287, 565-88 (2009) CrossRef
  8. Amor, Hadj: Sana: Regularity of the rotation number for the one-dimensional time-continuous Schr?dinger equation. Math. Phys. Anal. Geom 15-4, 331-42 (2012) CrossRef
  9. Herman, Michael-R.: Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnol \(\prime \) d et de Moser sur le tore de dimension 2. Commentarii Mathematici Helvetici 58-3, 453-02 (1983) CrossRef
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  11. Last, Y.: A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants. Commun. Math. Phys. 151, 183-92 (1993) CrossRef
  
• 作者单位：Sana Hadj Amor (1)
  
  1. Ecole sup?rieure des sciences et technologies de Hammam Sousse, Rue Lamine Elabbassi, 4011, Hammam Sousse, Tunisia
  
• ISSN：1572-9656

文摘

We show that the rotation number ρ(E) of the skew-product system \((\omega , A_{E}):(\theta , y)\in \mathbb T^{d}\times \mathbb R^{2}\mapsto (\theta +\omega , A(E, \theta )y)\in \mathbb T^{d}\times \mathbb R^{2},\) where ω is Diophantine, \(E\in \mathbb R\) and \(A(E, \theta )=A(E)+F(E, \theta )\in SL(2, \mathbb R)\) is homotopic to the identity, is absolutely continuous under a smallness condition on F. We deduce this fact from results already obtained on the reducibility of this skew product and on the regularity properties of its rotation number using perturbation theory of K.A.M. type.