Rigidity results for quasiperiodic *SL(2, R)*-cocycles

Pages: 479 - 510,
Issue 4,
October
2009 doi:10.3934/jmd.2009.3.479

Bassam Fayad - CNRS LAGA, Université Paris 13, 93430 Villetaneuse, France (email)

Raphaël Krikorian - Laboratoire de Probabilités et Modèles aléatoires, Université Pierre et Marie Curie, Boite courrier 188, 75252–Paris Cedex 05, France (email)

Abstract:
In this paper we introduce a new technique that allows us to investigate reducibility properties of
smooth *SL(2, R)*-cocycles over irrational rotations of the circle beyond the usual Diophantine conditions on these rotations.

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.

Keywords: Quasiperiodic cocycles, reducibility, KAM theory, Diophantine and Liouvillian
numbers.

Mathematics Subject Classification: Primary: 34C20; Secondary: 37Cxx.

Received: February 2009;
Revised:
September 2009;
Available Online: January 2010.