Katsura-Exel-Pardo self-similar actions, Putnam's binary factors and their limit spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918073112461312 |
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| author | Hume, Jeremy B. Whittaker, Michael F. |
| author_facet | Hume, Jeremy B. Whittaker, Michael F. |
| contents | We show that the dynamical system associated by Putnam to a pair of graph embeddings is identical to the shift map on the limit space of a self-similar groupoid action on a graph. Moreover, performing a certain out-split on said graph gives rise to a Katsura-Exel-Pardo groupoid action on the out-split graph whose associated limit space dynamical system is conjugate to the previous one. We characterise the self-similar properties of these groupoids in terms of properties of their defining data, two matrices $A$, $B$. We prove a large class of the associated limit spaces are bundles of circles and points which fibre over a totally disconnected space, and the dynamics restricted to each circle is of the form $z\to z^{n}$. Moreover, we find a planar embedding of these spaces, thereby answering a question Putnam posed in his paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_19863 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Katsura-Exel-Pardo self-similar actions, Putnam's binary factors and their limit spaces Hume, Jeremy B. Whittaker, Michael F. Dynamical Systems Operator Algebras 37B05, 37B10 (Primary), 37A55 (Secondary) We show that the dynamical system associated by Putnam to a pair of graph embeddings is identical to the shift map on the limit space of a self-similar groupoid action on a graph. Moreover, performing a certain out-split on said graph gives rise to a Katsura-Exel-Pardo groupoid action on the out-split graph whose associated limit space dynamical system is conjugate to the previous one. We characterise the self-similar properties of these groupoids in terms of properties of their defining data, two matrices $A$, $B$. We prove a large class of the associated limit spaces are bundles of circles and points which fibre over a totally disconnected space, and the dynamics restricted to each circle is of the form $z\to z^{n}$. Moreover, we find a planar embedding of these spaces, thereby answering a question Putnam posed in his paper. |
| title | Katsura-Exel-Pardo self-similar actions, Putnam's binary factors and their limit spaces |
| topic | Dynamical Systems Operator Algebras 37B05, 37B10 (Primary), 37A55 (Secondary) |
| url | https://arxiv.org/abs/2405.19863 |