Almost Everywhere Positivity of the Lyapunov Exponent for the Doubling Map

TL;DR

This paper demonstrates that for a class of Schrödinger operators with potentials generated by the doubling map, the Lyapunov exponent is positive almost everywhere, implying the absence of absolutely continuous spectrum.

Contribution

It establishes the almost everywhere positivity of the Lyapunov exponent for Schrödinger operators with doubling map potentials, linking ergodic dynamics to spectral properties.

Findings

Lyapunov exponent is positive almost everywhere.

Absolutely continuous spectrum is almost surely empty.

Potential realization as restrictions of non-deterministic operators.

Abstract

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.