Georgia Institute of Technology College of Sciences

A 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.