Positive Lyapunov exponent for some Schrödinger cocycles over strongly expanding circle endomorphisms

Abstract

We show that for a large class of potential functions and big coupling constant $\lambda$ the Schrödinger 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 $b\ge {\lambda }^3$.