Uniform positivity of Lyapunov exponents for anti Hölder potentials

Abstract

We consider Schrödinger operators with dynamically defined potentials that are anti $α$-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 $d\geq 1$. In particular, the application leads to uniformly positive Lyapunov exponents with sufficiently large coupling constants.