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We show that for a large class of potential functions and big couplingconstant $\lambda$ the Schr\"odinger cocycle over the expanding map $x\mapstobx ~( \text{mod} 1)$ on $\mathbb{T}$ has a Lyapunov exponent $>(\log\lambda)/4$for all energies, provided that the integer $b\geq \lambda^3$.

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