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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2205.01537 |
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| _version_ | 1866909483252318208 |
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| author | Putnam, Ian F. Treviño, Rodrigo |
| author_facet | Putnam, Ian F. Treviño, Rodrigo |
| contents | In [LT16], Kathryn Lindsey and the second author constructed a translation surface from a bi-infinite Bratteli diagram. We continue an investigation into these surfaces. The construction given in [LT16] was essentially combinatorial. Here, we provide explicit links between the path space of the Bratteli diagram and the surface, including various intermediate topological spaces. This allows us to relate the $C^{*}$-algebras associated with tail equivalence on the Bratteli diagram and the foliation of the surface, under some mild hypotheses. This also allows us to relate the K-theory of the $C^{*}$-algebras involved. We also treat the case of finite genus surfaces in some detail, where the process of Rauzy-Veech induction (and its inverse) provide an explicit construction of the Bratteli diagrams involved. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_01537 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Bratteli diagrams, translation flows and their $C^*$-algebras Putnam, Ian F. Treviño, Rodrigo Operator Algebras Dynamical Systems 19K14, 37C86, In [LT16], Kathryn Lindsey and the second author constructed a translation surface from a bi-infinite Bratteli diagram. We continue an investigation into these surfaces. The construction given in [LT16] was essentially combinatorial. Here, we provide explicit links between the path space of the Bratteli diagram and the surface, including various intermediate topological spaces. This allows us to relate the $C^{*}$-algebras associated with tail equivalence on the Bratteli diagram and the foliation of the surface, under some mild hypotheses. This also allows us to relate the K-theory of the $C^{*}$-algebras involved. We also treat the case of finite genus surfaces in some detail, where the process of Rauzy-Veech induction (and its inverse) provide an explicit construction of the Bratteli diagrams involved. |
| title | Bratteli diagrams, translation flows and their $C^*$-algebras |
| topic | Operator Algebras Dynamical Systems 19K14, 37C86, |
| url | https://arxiv.org/abs/2205.01537 |