Uniform positivity of Lyapunov exponents for anti Hölder potentials

Abstract

We consider Schrödinger operators with dynamically defined potentials that are anti $\alpha$-Hölder and generated by hyperbolic dynamics. We prove an anti Lipschitz estimate on functions related to the orbit of ${\vec{e}}_1$ under an associated projectivized cocycle. After, we apply the estimate on subshifts of finite type and maps with one expanding direction on ${\mathbb{T}}^d$ where $d\ge 1$. In particular, the application leads to uniformly positive Lyapunov exponents with sufficiently large coupling constants.