Aperiodically ordered structures are of considerable interest as models of quasicrystals, which have been known to exist in the physical world since 1982. In this talk, we describe how to construct sequences and tilings with aperiodic order using self-similar methods. We then discuss how (and why) it makes sense to embed them in a topological space (the "hull") and consider the action of translation.

The dynamical system that results from this process can be analyzed in many ways. In our talk we focus on the analytic, and how three different but related matrices (the third of which is new) can be used to obtain results on the spectrum.