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arXiv · 2112.07652

Self-similarity and limit spaces of substitution tiling semigroups

James J. Walton·Michael F. Whittaker
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Abstract

We show that Kellendonk's tiling semigroup of an FLC 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.

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James J. Walton, Michael F. Whittaker. 2021-12-14. Self-similarity and limit spaces of substitution tiling semigroups. https://arxiv.org/abs/2112.07652

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