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Rigidity results for quasiperiodic SL(2, R)-cocycles
| 1. | CNRS LAGA, Université Paris 13, 93430 Villetaneuse, France |
| 2. | 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 [5].
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 [1] 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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