Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles

Claire Chavaudret

Bulletin de la Société Mathématique de France (2013)

  • Volume: 141, Issue: 1, page 47-106
  • ISSN: 0037-9484

Abstract

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This article is about almost reducibility of quasi-periodic cocycles with a diophantine frequency which are sufficiently close to a constant. Generalizing previous works by L.H. Eliasson, we show a strong version of almost reducibility for analytic and Gevrey cocycles, that is to say, almost reducibility where the change of variables is in an analytic or Gevrey class which is independent of how close to a constant the initial cocycle is conjugated. This implies a result of density, or quasi-density, of reducible cocycles near a constant. Some algebraic structure can also be preserved, by doubling the period if needed.

How to cite

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Chavaudret, Claire. "Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles." Bulletin de la Société Mathématique de France 141.1 (2013): 47-106. <http://eudml.org/doc/272565>.

@article{Chavaudret2013,
abstract = {This article is about almost reducibility of quasi-periodic cocycles with a diophantine frequency which are sufficiently close to a constant. Generalizing previous works by L.H. Eliasson, we show a strong version of almost reducibility for analytic and Gevrey cocycles, that is to say, almost reducibility where the change of variables is in an analytic or Gevrey class which is independent of how close to a constant the initial cocycle is conjugated. This implies a result of density, or quasi-density, of reducible cocycles near a constant. Some algebraic structure can also be preserved, by doubling the period if needed.},
author = {Chavaudret, Claire},
journal = {Bulletin de la Société Mathématique de France},
keywords = {small divisors; small denominators; quasiperiodic skew-product; quasiperiodic cocycles; Lyapunov exponent; Floquet theory},
language = {eng},
number = {1},
pages = {47-106},
publisher = {Société mathématique de France},
title = {Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles},
url = {http://eudml.org/doc/272565},
volume = {141},
year = {2013},
}

TY - JOUR
AU - Chavaudret, Claire
TI - Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles
JO - Bulletin de la Société Mathématique de France
PY - 2013
PB - Société mathématique de France
VL - 141
IS - 1
SP - 47
EP - 106
AB - This article is about almost reducibility of quasi-periodic cocycles with a diophantine frequency which are sufficiently close to a constant. Generalizing previous works by L.H. Eliasson, we show a strong version of almost reducibility for analytic and Gevrey cocycles, that is to say, almost reducibility where the change of variables is in an analytic or Gevrey class which is independent of how close to a constant the initial cocycle is conjugated. This implies a result of density, or quasi-density, of reducible cocycles near a constant. Some algebraic structure can also be preserved, by doubling the period if needed.
LA - eng
KW - small divisors; small denominators; quasiperiodic skew-product; quasiperiodic cocycles; Lyapunov exponent; Floquet theory
UR - http://eudml.org/doc/272565
ER -

References

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  1. [1] A. Avila & R. Krikorian – « Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles », Ann. of Math.164 (2006), p. 911–940. Zbl1138.47033MR2259248
  2. [2] C. Chavaudret – « Reducibility of quasi-periodic cocycles in linear lie groups », Ergod. Theory and Dyn. Syst.31 (2011), p. 741–769. 
  3. [3] L. H. Eliasson – « Floquet solutions for the 1 -dimensional quasi-periodic Schrödinger equation », Comm. Math. Phys.146 (1992), p. 447–482. Zbl0753.34055MR1167299
  4. [4] —, « Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum », Acta Math.179 (1997), p. 153–196. Zbl0908.34072MR1607554
  5. [5] —, « Almost reducibility of linear quasi-periodic systems », in Smooth ergodic theory and its applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., 2001, p. 679–705. Zbl1015.34028MR1858550
  6. [6] S. Hadj Amor – « Opérateur de Schrödinger quasi-périodique unidimensionnel », thèse de doctorat, Université Paris 7, 2006. 
  7. [7] R. Krikorian – « Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts », Astérisque 259 (1999). 
  8. [8] J.-P. Marco & D. Sauzin – « Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems », Publ. Math. I.H.É.S. 96 (2002), p. 199–275. Zbl1086.37031MR1986314

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