Uniform positivity of the Lyapunov exponent for monotone potentials generated by the doubling map

TL;DR

This paper proves that for a class of monotone potentials generated by the doubling map, the Lyapunov exponent remains uniformly positive across all energies when the coupling constant is sufficiently large, indicating strong localization properties.

Contribution

It establishes uniform positivity of the Lyapunov exponent for monotone potentials generated by the doubling map, a result not previously known for this class of systems.

Findings

Lyapunov exponent is uniformly positive for large coupling

Results hold for all energies in the spectrum

Applicable to potentials generated by the doubling map

Abstract

We show that for any doubling map generated $C^1$ monotone potential with derivative uniformly bounded away from zero, the Lyapunov exponent of the associated Schr\"odinger operators is uniformly positive for all energies provided the coupling constant is large.