Geodesic
Self-similarity and limit spaces of substitution tiling semigroups
James Walton1 orcid ; Michael F. Whittaker2 orcid
1University of Nottingham, Nottingham, UK
2University of Glasgow, Glasgow, UK
Groups, geometry, and dynamics, Tome 18 (2024) no. 4, pp. 1201-1231

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DOI

We show that Kellendonk’s tiling semigroup of a finite local complexity substitution tiling is self-similar, in the sense of Bartholdi, Grigorchuk and Nekrashevych. We extend the notion of the limit space of a self-similar group to the setting of self-similar semigroups, and show that it is homeomorphic to the Anderson–Putnam complex for such substitution tilings, with natural self-map induced by the substitution. Thus, the inverse limit of the limit space, given by the limit solenoid of the self-similar semigroup, is homeomorphic to the translational hull of the tiling.
DOI : 10.4171/ggd/807
Classification : 37B52, 20M18, 52C22
Mots-clés : aperiodic tilings, self-similar, semigroups, tiling dynamics

James Walton  1   ; Michael F. Whittaker  2

1 University of Nottingham, Nottingham, UK
2 University of Glasgow, Glasgow, UK 
James Walton; Michael F. Whittaker. Self-similarity and limit spaces of substitution tiling semigroups. Groups, geometry, and dynamics, Tome 18 (2024) no. 4, pp. 1201-1231. doi: 10.4171/ggd/807
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