Multiscale Analysis for Ergodic Schr\"odinger operators and positivity of Lyapunov exponents

A variant of multiscale analysis for ergodic Schr\"odinger operators is developed. This enables us to prove positivity of Lyapunov exponents given initial scale estimates and an initial Wegner estimate. This is then applied to high dimensional skew-shifts at small coupling, where initial conditions are checked using the Pastur--Figotin formalism. Furthermore, it is shown that for potentials generated by the doubling map one has positive Lyapunov exponent except in a superpolynomially small set.