Dynamics and topology of the Hat family of tilings
Abstract
The recently discovered Hat tiling admits a 4dimensional family of shape deformations, including the 1parameter family already known to yield alternate monotiles. The continuous hulls resulting from these tilings are all topologically conjugate dynamical systems, and hence have the same dynamics and topology. We construct and analyze a selfsimilar element of this family called the CAP tiling, and we use it to derive properties of the entire family. The CAP tiling has purepoint dynamical spectrum, which we compute explicitly, and comes from a natural cutandproject scheme with 2dimensional Euclidean internal space. All other members of the Hat family, in particular the original version constructed from 306090 right triangles, are obtained via small modifications of the projection from this cutandproject scheme.
 Publication:

arXiv eprints
 Pub Date:
 May 2023
 DOI:
 10.48550/arXiv.2305.05639
 arXiv:
 arXiv:2305.05639
 Bibcode:
 2023arXiv230505639B
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Metric Geometry;
 52C20;
 37D40;
 55N05;
 52C23
 EPrint:
 30 pages, 10 figures, 3 tables