Indistinguishable asymptotic pairs and multidimensional Sturmian configurations
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866929685091319808 |
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| author | Barbieri, Sebastián Labbé, Sébastien |
| author_facet | Barbieri, Sebastián Labbé, Sébastien |
| contents | Two asymptotic configurations on a full $\mathbb{Z}^d$-shift are indistinguishable if for every finite pattern the associated sets of occurrences in each configuration coincide up to a finitely supported permutation of $\mathbb{Z}^d$. We prove that indistinguishable asymptotic pairs satisfying a "flip condition" are characterized by their pattern complexity on finite connected supports. Furthermore, we prove that uniformly recurrent indistinguishable asymptotic pairs satisfying the flip condition are described by codimension-one (dimension of the internal space) cut and project schemes, which symbolically correspond to multidimensional Sturmian configurations. Together the two results provide a generalization to $\mathbb{Z}^d$ of the characterization of Sturmian sequences by their factor complexity $n+1$. Many open questions are raised by the current work and are listed in the introduction. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2204_06413 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Indistinguishable asymptotic pairs and multidimensional Sturmian configurations Barbieri, Sebastián Labbé, Sébastien Dynamical Systems Combinatorics Primary 37B10, Secondary 37C29, 52C23, 68R15 Two asymptotic configurations on a full $\mathbb{Z}^d$-shift are indistinguishable if for every finite pattern the associated sets of occurrences in each configuration coincide up to a finitely supported permutation of $\mathbb{Z}^d$. We prove that indistinguishable asymptotic pairs satisfying a "flip condition" are characterized by their pattern complexity on finite connected supports. Furthermore, we prove that uniformly recurrent indistinguishable asymptotic pairs satisfying the flip condition are described by codimension-one (dimension of the internal space) cut and project schemes, which symbolically correspond to multidimensional Sturmian configurations. Together the two results provide a generalization to $\mathbb{Z}^d$ of the characterization of Sturmian sequences by their factor complexity $n+1$. Many open questions are raised by the current work and are listed in the introduction. |
| title | Indistinguishable asymptotic pairs and multidimensional Sturmian configurations |
| topic | Dynamical Systems Combinatorics Primary 37B10, Secondary 37C29, 52C23, 68R15 |
| url | https://arxiv.org/abs/2204.06413 |