Abstract: The recently discovered Hat is an aperiodic monotile for the Euclidean plane: Together with rotated and reflected copies, it can tile the plane, but only aperiodically. It is of interest to understand what kind of long-range order emerges, and how this compare with previous examples, such as the Taylor--Socolar monotile. To do so, methods from topology, dynamics and harmonic analysis are combined to show that the Hat tiling induces a dynamical system that is topologically conjugate to a primitive inflation tiling, the CAP, from which it then inherits a quasiperiodic structure with pure point spectrum and continuous eigenfunctions. In particular, it can be understood via a projection from four dimensions. A similar structure is present for the even more recent Spectre tiling, which is another aperiodic monotile for the plane, this time without needing a reflected version. This is joint work with Franz Gaehler and Lorenzo Sadun.


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