Almost Everywhere Positivity of the Lyapunov Exponent for the Doubling Map

Details Almost Everywhere Positivity of the Lyapunov Exponent for the Doubling Map Almost Everywhere Positivity of the Lyapunov Exponent for the Doubling Map

2004-05-25

arxiv.org/abs/math-ph/0405061v1

Physics

Mathematical Physics

4 pages

Scientific paper

10.1007/s00220-004-1261-x

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.

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