## 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

Zbl 0957.37016
Full Text:

### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.