Absolute Continuity of the Rotation Number for Quasi-Periodic CO-Cycles in $SL(2, \mathbb {R})$

Abstract

We show that the rotation number ρ(E) of the skew-product system $(\omega , A_{E}):(\theta , y)\in \mathbb T^{d}\times \mathbb R^{2}\mapsto (\theta +\omega , A(E, \theta )y)\in \mathbb T^{d}\times \mathbb R^{2},$ where ω is Diophantine, $E\in \mathbb R$ and $A(E, \theta )=A(E)+F(E, \theta )\in SL(2, \mathbb R)$ is homotopic to the identity, is absolutely continuous under a smallness condition on F. We deduce this fact from results already obtained on the reducibility of this skew product and on the regularity properties of its rotation number using perturbation theory of K.A.M. type.