We discuss the dynamics of Schr¨odinger cocycles over Diophantine rotations of the circle in the case of non-perturbatively small analytic potential. In such systems, the dynamics is conjugate to a constant for typical energies in the spectrum, but often one finds also a generic set of energies for which the behavior is more erratic. Something remains however in the form of a (possibly divergent) Fourier series of a “would be conjugacy”. We show how a careful analysis of such Fourier series leads to detailed description of the dynamics. This allows us to obtain sharp estimates on some characteristics of the related operators. Co-author: Svetlana Jitomirskay