  To understand an aperiodic tiling (or a quasicrystal modeled on an aperiodic
tiling), we construct a space of similar tilings, on which the group of
translations acts naturally. This space is then an (abstract) dynamical system.
Dynamical properties of the space (such as mixing, or the spectrum of the
translation operator) are closely related to bulk properties of the individual
tilings (such as the diffraction pattern). The topology of the space of
tilings, particularly the Cech cohomology, gives information on how the
original tiling can be deformed. Tiling spaces can be constructed as inverse
limits of branched manifolds.
