LAUSR

We initiate the study of Schrödinger operators with ergodic potentials defined over circle map dynamics, in particular over circle diffeomorphisms. For analytic circle diffeomorphisms and a set of rotation numbers satisfying Yoccoz’sH arithmetic condition, we discuss an extension of Avila’s global theory. We also prove a sharp Gordon-type theorem which implies that for every C1+BV circle diffeomorphism, with a Liouville rotation number and an invariant measure μ, for μ-almost all x ∈ T1, the corresponding Schrödinger operator has purely continuous spectrum for every Hölder continuous potential V.