AA << generalize the notion of (geometric) substitution rule to obtain overlapping substitutions. For such a substitution, the substitution matrix may have non-integer entries. (They) 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. >>
AA << give a sufficient condition for an overlapping substitution to be consistent, that is, no such contradiction will emerge. (They) also give many one-dimensional examples of overlapping substitution. (AA) finish by mentioning a construction of overlapping substitutions from Delone multi-sets with inflation symmetry. >>️
Shigeki Akiyama, Yasushi Nagai, Shu-Qin Zhang. Overlapping substitutions and tilings. arXiv: 2407.18666v2 [math.CO]. Aug 30, 2024.
Also
keyword 'tile', in FonT
keyword 'overlap', 'overlapping', in FonT
Keywords: gst, tiles, overlap, overlapping
