  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.
