Self-similarity and limit spaces of substitution tiling semigroups

Walton, James; Whittaker, Michael F.

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.

Citation

Walton, J., & Whittaker, M. F. (2024). Self-similarity and limit spaces of substitution tiling semigroups. Groups, Geometry, and Dynamics, 18(4), 1201-1231. https://doi.org/10.4171/ggd/807

• Journal Article Type: Article
• Acceptance Date: Apr 6, 2024
• Online Publication Date: Jul 11, 2024
• Publication Date: Jul 11, 2024
• Deposit Date: Apr 25, 2024
• Publicly Available Date: Apr 25, 2024
• Journal: Groups, Geometry, and Dynamics
• Print ISSN: 1661-7207
• Electronic ISSN: 1661-7215
• Publisher: European Mathematical Society
• Peer Reviewed: Peer Reviewed
• Volume: 18
• Issue: 4
• Pages: 1201-1231
• DOI: https://doi.org/10.4171/ggd/807
• Keywords: aperiodic tilings; self-similar; semigroups; tiling dynamics
• Public URL: https://nottingham-repository.worktribe.com/output/34105285
• Publisher URL: https://ems.press/journals/ggd/articles/14297982