Positive Lyapunov exponent for some Schr\"odinger cocycles over strongly expanding circle endomorphisms

We show that for a large class of potential functions and big coupling constant $\lambda$ the Schr"odinger cocycle over the expanding map $x\mapsto bx ~( $\text{mod}$ 1)$on$$\mathbb{T}$$hasaLyapunovexponent$>(\log\lambda)/4$forallenergies,providedthattheinteger$b\geq \lambda^3$.