Rigidity of reducibility of Gevrey quasi-periodic cocycles on $U(n)$

Hou, Xuanji; Popov, Georgi

doi:10.24033/bsmf.2705

Rigidity of reducibility of Gevrey quasi-periodic cocycles on

U (n)

[Rigidité de réductibilité des cocycles quasi-périodiques de Gevrey sur

U (n)

]

Hou, Xuanji ; Popov, Georgi

Bulletin de la Société Mathématique de France, Tome 144 (2016) no. 1, pp. 1-52

Abstract (VO)
Résumé

We consider the reducibility problem of cocycles $(α, A)$ on $𝕋^{d} \times U (n)$ in Gevrey classes, where $α$ is a Diophantine vector. We prove that, if a Gevrey cocycle is conjugated to a constant cocycle $(α, C)$ by a suitable measurable conjugacy $(0, B)$ , then for almost all $C$ it can be conjugated to $(α, C)$ in the same Gevrey class, provided that $A$ is sufficiently close to a constant. If $B$ is continuous we obtain that it is Gevrey smooth. We consider as well the global problem of reducibility in Gevrey classes when $d = 1$ .

On considère le problème de la réductibilité de cocycles $(α, A)$ sur $𝕋^{d} \times U (n)$ dans les classes de Gevrey, où $α$ est Diophantien. Si $A$ est proche d'une constante et le Gevrey cocycle $(α, A)$ est conjuqué au cocycle constant $(α, C)$ par une conjugaison mesurable $(0, B)$ , on montre que pour presque tous $C$ le cocycle peut êtrte conjuguer à $(α, C)$ dans la même classe de Gevrey . Si $B$ est continue on obtient qu'elle est Gevrey. On considère aussi le problème de la réductibilité globale dans les classes de Gevrey dans le cas où $d = 1$ .

Publié le : 2016-07-21

MR Zbl

DOI : 10.24033/bsmf.2705

Keywords: reducibility of quasi-periodic cocycles, Gevrey classes

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     author = {Hou, Xuanji and Popov, Georgi},
     title = {Rigidity of reducibility of {Gevrey} quasi-periodic cocycles on~$U(n)$},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {1--52},
     year = {2016},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {144},
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     mrnumber = {3481260},
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}

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Hou, Xuanji; Popov, Georgi. Rigidity of reducibility of Gevrey quasi-periodic cocycles on $U(n)$. Bulletin de la Société Mathématique de France, Tome 144 (2016) no. 1, pp. 1-52. doi: 10.24033/bsmf.2705

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