1. Overlapping substitutions and tilings
- Author
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Akiyama, Shigeki, Nagai, Yasushi, and Zhang, Shu-Qin
- Subjects
Mathematics - Combinatorics ,Mathematics - Dynamical Systems ,Mathematics - Number Theory - Abstract
Inspired by \cite{DBZ},we generalize the notion of (geometric) substitution rule to obtain overlapping substitutions. For such a substitution, the substitution matrix may have non-integer entries. We give the meaning of such a matrix by showing that the right Perron--Frobenius eigenvector gives the patch frequency of the resulting tiling. We also show that the expansion constant is an algebraic integer under mild conditions. In general, overlapping substitutions may yield a patch with contradictory overlaps of tiles, even if it is locally consistent. We give a sufficient condition for an overlapping substitution to be consistent, that is, no such contradiction will emerge. We also give many one-dimensional examples of overlapping substitution. We finish by mentioning a construction of overlapping substitutions from Delone multi-sets with inflation symmetry.
- Published
- 2024