Hölder Continuity of the Rotation Number for Quasi-Periodic Co-Cycles in

We prove two results on the rotation number of the skew-product system where ω is Diophantine and is homotopic to the identity. On the one hand, we prove that this function has the behavior of a Hölder function. On the other, we show that the length of the gaps has a sub-exponential estimate which depends on its label given by the gap-labeling theorem. We give also an estimate of the complement of the spectrum. These results are obtained by studying the reducibility of the quasi-periodic co-cycle ( ω , A ).