# Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles

Bulletin de la Société Mathématique de France (2013)

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

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topChavaudret, 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

top- [1] A. Avila & R. Krikorian – « Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles », Ann. of Math.164 (2006), p. 911–940. Zbl1138.47033MR2259248
- [2] C. Chavaudret – « Reducibility of quasi-periodic cocycles in linear lie groups », Ergod. Theory and Dyn. Syst.31 (2011), p. 741–769.
- [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] —, « Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum », Acta Math.179 (1997), p. 153–196. Zbl0908.34072MR1607554
- [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] S. Hadj Amor – « Opérateur de Schrödinger quasi-périodique unidimensionnel », thèse de doctorat, Université Paris 7, 2006.
- [7] R. Krikorian – « Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts », Astérisque 259 (1999).
- [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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