Almost Everywhere Positivity of the Lyapunov Exponent for the Doubling   Map
  We show that discrete one-dimensional Schr\"odinger operators on the half-line with ergodic potentials generated by the doubling map on the circle, $V_\theta(n) = f(2^n \theta)$, may be realized as the half-line restrictions of a non-deterministic family of whole-line operators. As a consequence, the Lyapunov exponent is almost everywhere positive and the absolutely continuous spectrum is almost surely empty. 
David Damanik (Caltech), Rowan Killip (UCLA)