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

Abstract We show that for any doubling map generated $$C^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> monotone potential with derivative uniformly bounded away from zero and infinity, the Lyapunov exponent of the associated Schrödinger operators is bounded below by $$\log {\lambda }-C$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>log</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>-</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> for all energies, where C depends only on the potential. In particular, it answers an open question [D, Problem 5] raised by D. Damanik.