On the cohomology of homshifts
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908604789948416 |
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| author | Chandgotia, Nishant Gangloff, Silvère de Menibus, Benjamin Hellouin Oprocha, Piotr |
| author_facet | Chandgotia, Nishant Gangloff, Silvère de Menibus, Benjamin Hellouin Oprocha, Piotr |
| contents | We study the cohomology of symbolic dynamical systems called homshifts: they are the nearest-neighbour $\mathbb{Z}^d$ shifts of finite type whose adjacency rules are the same in every direction. Building on the work of Klaus Schmidt (Pacific J. Math. 170 (1995), no.1, 237-269) we give a necessary and sufficient condition for homshifts to be cohomological trivial. This condition is expressed in terms of the topology of a natural simplicial complex arising from the shift space which can be analyzed in many natural cases. However, we prove that in general, cohomological triviality is algorithmically undecidable for homshifts. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_11969 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the cohomology of homshifts Chandgotia, Nishant Gangloff, Silvère de Menibus, Benjamin Hellouin Oprocha, Piotr Dynamical Systems 37B51 05C25 37A20 We study the cohomology of symbolic dynamical systems called homshifts: they are the nearest-neighbour $\mathbb{Z}^d$ shifts of finite type whose adjacency rules are the same in every direction. Building on the work of Klaus Schmidt (Pacific J. Math. 170 (1995), no.1, 237-269) we give a necessary and sufficient condition for homshifts to be cohomological trivial. This condition is expressed in terms of the topology of a natural simplicial complex arising from the shift space which can be analyzed in many natural cases. However, we prove that in general, cohomological triviality is algorithmically undecidable for homshifts. |
| title | On the cohomology of homshifts |
| topic | Dynamical Systems 37B51 05C25 37A20 |
| url | https://arxiv.org/abs/2510.11969 |