The authors are interested in the Anderson localization phenomenon and write the present manuscript as part of a series of papers. Here they consider the one-dimensional discrete Schrödinger operators \(H_\omega\) in \(\ell^2 (\mathbb{Z} )\), given explicitly by \(H_\omega (n) = \psi(n+1) + \psi(n-1) + V_\omega(n) \psi(n)\). The potentials \(V_\omega: \mathbb{Z} \rightarrow \mathbb{R}\) are defined by \(V_\omega (n) = f (T^n \omega)\) for \(\omega \in \Omega\) and \(n \in \mathbb{Z}\). It is assumed that \(\Omega\) is a compact metric space, \(T: \Omega \rightarrow \Omega\) is a homeomorphism, and \(f: \Omega \rightarrow \mathbb{R}\) is continuous.
A generalization that incorporates most of the systems under study is the following: \((\Omega, T)\) is taken to be a subshift of finite type with a \(T\)-ergodic measure \(\mu\) entirely supported on \(\Omega\). It is also assumed that \(\mu\) allows a local product structure and that \(f:\Omega \rightarrow \mathbb{R}\) is a non-constant \(\alpha\)-Hölder continuous function for \(0 < \alpha \le 1\).
The authors are mainly interested in the Lyapunov exponent of the Schrödinger cocycle of the basic operator, and in particular in studying the set \(\mathcal{Z}_f\) where the Lyapunov exponent is zero. Their main result is that if \(T\) has a fixed point and \(f\) is Hölder continuous and non-constant, then \(\mathcal{Z}_f\) is discrete.