On Schrödinger operators with dynamically defined potentials

The purpose of this article is to review some of the recent work on the operator
$$ (H_\psi)_n=-\psi_{n-1}-\psi_{n+1}+\lambda V(T^n x)\psi_n $$
on $\ell^2(\mathbb Z)$, where $T\colon X\to X$ is an ergodic transformation on $(X,\nu)$ and $V$ is a real-valued function. $\lambda$ is a real parameter called coupling constant. Typically, $X=\mathbb T^d=(\mathbb R/\mathbb Z)^d$ with Lebesgue measure, and $V$ will be a trigonometric polynomial or analytic. We shall focus on our earlier papers, as well as other work which was obtained jointly with Jean Bourgain. Our goal is to explain some of the methods and results from these references. Some of the material in this paper has not appeared elsewhere in print.