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

TL;DR

This paper demonstrates that for certain potential functions and large coupling constants, the Schrödinger cocycle over an expanding circle map has a positive Lyapunov exponent exceeding a specific bound, under certain conditions on the parameters.

Contribution

It establishes a lower bound on the Lyapunov exponent for Schrödinger cocycles over expanding circle maps for a broad class of potentials and large coupling constants.

Findings

Lyapunov exponent exceeds (\log\lambda)/4 for large \lambda

Valid for all energies under specified conditions

Applicable to a large class of potential functions

Abstract

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 $b\geq \lambda^3$.