The positivity of the Lyapunov exponent and the absence of the absolutely continuous spectrum for the almost-Mathieu equation

This paper contains the rigorous proof of the formulated by Andre and Aubry following statement: the Cauchy solutions of the discrete Schrödinger equation with the potential qn=g cos(2πnθ+φ) grow exponentially for every irrational θ, g>1 and almost every φε[0,2π). According to known this fact implies the absence of the absolutely continuous component of the spectrum for the corresponding operator.