  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 $\mathbb{T}$ has a Lyapunov exponent $>(\log\lambda)/4$
for all energies, provided that the integer $b\geq \lambda^3$.
