Almost Everywhere Positivity of the Lyapunov Exponent for the Doubling Map

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We show that discrete one-dimensional Schrödinger operators on the half-line with ergodic potentials generated by the doubling map on the circle, Vθ(n) = f(2ⁿ θ), 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.