  We review ergodic properties of one-dimensional incommensurate bilayer
materials, especially the convergence of the density of states in the
large-volume limit, from the perspective of the theory of ergodic Schr\"odinger
operators. More precisely, we first provide a short introduction to ergodic
Schr\"odinger operators as a unifying concept in spectral theory at a level
accessible for nonspecialists. We then present two natural tight-binding models
of incommensurate bilayer materials in one dimension and prove convergence of
their density of states measures in the large volume limit using ideas from the
theory of ergodic Schr\"odinger operators.
