Spectral Theory of Schrodinger Operators over Circle Diffeomorphisms

We initiate the study of Schrodinger 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's H arithmetic condition, we discuss an extension of Avila's global theory. We also give an abstract version and a short proof of a sharp Gordon-type theorem on the absence of eigenvalues for general potentials with repetitions. Coupled with the dynamical analysis, we obtain that, for every C1+BV circle diffeomorphism, with a super Liouville rotation number and an invariant measure mu, and for mu-almost all x is an element of T-1, the corresponding Schrodinger operator has purely continuous spectrum for every Holder continuous potential V.