Uniform Positivity of the Lyapunov Exponent for Monotone Potentials Generated by the Doubling Map

We show that for any doubling map generated \(C^1\) $C^1$ monotone potential with derivative uniformly bounded away from zero and infinity, the Lyapunov exponent of the associated Schrödinger operators is bounded below by \(\log {\lambda }-C\) $log\lambda -C$ for all energies, where C depends only on the potential. In particular, it answers an open question [D, Problem 5] raised by D. Damanik.