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We show that discrete one-dimensional Schr\"odinger operators on thehalf-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 ofa non-deterministic family of whole-line operators. As a consequence, theLyapunov exponent is almost everywhere positive and the absolutely continuousspectrum is almost surely empty.Comment: 4 page

Almost Everywhere Positivity of the Lyapunov Exponent for the Doubling Map