Almost Localization and Almost Reducibility

We develop a quantitative version of Aubry duality and use it to obtain several sharp estimates for the dynamics of Schrödinger cocycles associated to a non-perturbatively small analytic potential and Diophantine frequency. In particular, we establish the full version of Eliasson’s reducibility theory in this regime (our approach actually leads to improvements even in the perturbative regime: we are able to show, for all energies, “almost reducibility” in some band of analyticity). We also prove 1/2-Hölder continuity of the integrated density of states. For the almost Mathieu operator, our results hold through the entire regime of sub-critical coupling and imply also the dry version of the Ten Martini Problem for the concerned parameters.