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Almost everywhere reducibility of quasi-periodic fibered flows with values in compact groups. (Réductibilité presque partout des flots fibrés quasi-périodiques à valeurs dans des groupes compacts.) (French) Zbl 1098.37510

Summary: Let us be given a compact “semisimple” Lie group \(G\) with Lie algebra \(g\), a regular element \(A\in g\), a bounded interval \(\Lambda\subset\mathbb R\) and a diophantine vector \(\omega\in\mathbb R^d\); then if \(F \in C^\omega(\mathbb R^d/\mathbb Z^d,g)\) is small enough, \(\omega\) meaning here “real analytic”, for Lebesgue-a.e. \(\lambda\in\Lambda\), the quasi-periodic system \(\lambda A+F((\omega_1/2\pi),\dots, (\omega_d/2\pi))\), with frequency vector \(\omega\), is Floquet-reducible modulo some finite covering depending only on the group \(G\). This theorem is a generalization of the one proved in the author’s monograph [Reducibility of skew-product systems with values in compact groups (French), Astérisque 259, Paris: SMF (1999; Zbl 0957.37016)].

MSC:

37C55 Periodic and quasi-periodic flows and diffeomorphisms
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations

Citations:

Zbl 0957.37016
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References:

[1] L.H. ELIASSON , Floquet solutions for the 1-Dimensional quasi-periodic Schrödinger equation , Comm. Math. Phys. 146, 1992 , p. 447-482. Article | MR 93d:34141 | Zbl 0753.34055 · Zbl 0753.34055
[2] L.H. ELIASSON , Ergodic Skew Systems on Td \times SO(3, R) Prépublication de l’ETH, Sept 1991 .
[3] L.H. ELIASSON , Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum , Acta Math. 179, 1997 , p. 153-196. MR 99k:47072 | Zbl 0908.34072 · Zbl 0908.34072
[4] L.H. ELIASSON , Perturbations of stable invariant tori for hamiltonian systems , Ann. Sc. Nor. Sup. di Pisa 4, 1988 , p. 115-147. Numdam | MR 91b:58060 | Zbl 0685.58024 · Zbl 0685.58024
[5] R. KRIKORIAN , Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts , à paraître dans Astérisque. Zbl 0957.37016 · Zbl 0957.37016
[6] A.S. PYARTLI , Diophantine approximations on submanifolds of euclidian spaces , Funkt. Anal. i. Priloz. 3, 1969 , p. 303-306. Zbl 0216.04401 · Zbl 0216.04401
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