A lattice in a real Euclidean space is the epitome of an infinite discrete system which is highly ordered. Long-range aperiodic order refers to the phenomenon of discrete systems of infinite extent which still have very evident order but are either partially or totally deficient in periodic symmetries. The order appears now in several ways, the most notable being the repetition of local structure (albeit aperiodically) and, most importantly, through the existence of a strong pure point component in the diffraction. Famous examples are the Penrose tilings, the Fibonacci substitution sequences, and the actual physical examples of quasi-crystalline materials.
Just as in the case of statistical mechanics, it has proven extremely useful to study aperiodic structures not just as individuals in isolation, but rather as members of some larger family of closely related aperiodic structures; for example, all those objects whose local structures are indistinguishable up to translation. Thereby arise dynamical systems, and again, just as in statistical mechanics, the spectral theory of the associated dynamical system provides a powerful method of exploring the underlying geometry of the original structure that led to it.
In this lecture we will introduce long-range aperiodic order and diffraction, and indicate with some new results just how effective dynamical systems can be in their study.