Abstract. 
The Penrose tiling is an example of an aperiodic tiling and its
vertex set is an example of an aperiodic point set (sometimes known as a
quasicrystal). There are higher rank dynamical systems associated with any
aperiodic tiling or point set, and in many cases they define a uniquely
ergodic action on a compact metric space. I will talk about the ergodic
theory of these systems. In particular, I will state the results of an
ongoing work with S. Schmieding on the deviations of ergodic averages of
such actions for point sets, where cohomology plays a big role. I'll relate
the results to the diffraction spectrum of the associated quasicrystals.