Weakly and Strongly Aperiodic Subshifts of Finite Type on Baumslag-Solitar Groups
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arXiv
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| Format: | Preprint |
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2020
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| author | Esnay, Solène J. Moutot, Etienne |
| author_facet | Esnay, Solène J. Moutot, Etienne |
| contents | We study the periodicity of subshifts of finite type (SFT) on Baumslag-Solitar groups. We show that for residually finite Baumslag-Solitar groups there exist both strongly and weakly-but-not-strongly aperiodic SFTs. In particular, this shows that unlike $\mathbb{Z}^2$, but like $\mathbb{Z}^3$, strong and weak aperiodic SFTs are different classes of SFTs in residually finite BS groups. More precisely, we prove that a weakly aperiodic SFT on BS(m,n) due to Aubrun and Kari is, in fact, strongly aperiodic on BS(1,n); and weakly but not strongly aperiodic on any other BS(m,n). In addition, we exhibit an SFT which is weakly but not strongly aperiodic on BS(1,n); and we show that there exists a strongly aperiodic SFT on BS(n,n). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2004_02534 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Weakly and Strongly Aperiodic Subshifts of Finite Type on Baumslag-Solitar Groups Esnay, Solène J. Moutot, Etienne Dynamical Systems Discrete Mathematics Group Theory We study the periodicity of subshifts of finite type (SFT) on Baumslag-Solitar groups. We show that for residually finite Baumslag-Solitar groups there exist both strongly and weakly-but-not-strongly aperiodic SFTs. In particular, this shows that unlike $\mathbb{Z}^2$, but like $\mathbb{Z}^3$, strong and weak aperiodic SFTs are different classes of SFTs in residually finite BS groups. More precisely, we prove that a weakly aperiodic SFT on BS(m,n) due to Aubrun and Kari is, in fact, strongly aperiodic on BS(1,n); and weakly but not strongly aperiodic on any other BS(m,n). In addition, we exhibit an SFT which is weakly but not strongly aperiodic on BS(1,n); and we show that there exists a strongly aperiodic SFT on BS(n,n). |
| title | Weakly and Strongly Aperiodic Subshifts of Finite Type on Baumslag-Solitar Groups |
| topic | Dynamical Systems Discrete Mathematics Group Theory |
| url | https://arxiv.org/abs/2004.02534 |