Rigidity results for quasiperiodic SL(2, R)-cocycles
CNRS LAGA, Université Paris 13, 93430 Villetaneuse, France
Laboratoire de Probabilités et Modèles aléatoires, Université Pierre et Marie Curie, Boite courrier 188, 75252–Paris Cedex 05
For any given irrational angle on the base, we show that if the cocycle has bounded fibered products and if its fibered rotation number belongs to a set of full measure $\Sigma(\a)$, then the matrix map can be perturbed in the $C^\infty$ topology to yield a $C^\infty$-reducible cocycle. Moreover, the cocycle itself is almost rotations-reducible in the sense that it can be conjugated arbitrarily close to a cocycle of rotations. If the rotation on the circle is of super-Liouville type, the same results hold if instead of having bounded products we only assume that the cocycle is $L^2$-conjugate to a cocycle of rotations.
When the base rotation is Diophantine, we show that if the cocycle is $L^2$-conjugate to a cocycle of rotations and if its fibered rotation number belongs to a set of full measure, then it is $C^\infty$-reducible. This extends a result proven in .
As an application, given any smooth SL(2, R)-cocycle over a irrational rotation of the circle, we show that it is possible to perturb the matrix map in the $C^\infty$ topology in such a way that the upper Lyapunov exponent becomes strictly positive. The latter result is generalized, based on different techniques, by Avila in  to quasiperiodic SL(2, R)-cocycles over higher-dimensional tori.
Also, in the course of the paper we give a quantitative version of a theorem by L. H. Eliasson, a proof of which is given in the Appendix. This motivates the introduction of a quite general KAM scheme allowing to treat bigger losses of derivatives for which we prove convergence.
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