Automaticity of uniformly recurrent substitutive sequences
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866912904543993856 |
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| author | Krawczyk, Elżbieta Müllner, Clemens |
| author_facet | Krawczyk, Elżbieta Müllner, Clemens |
| contents | We provide a complete characterisation of automaticity of uniformly recurrent substitutive sequences in terms of the incidence matrix of the return substitution of the underlying purely substitutive sequence. This resolves a recent question posed by Allouche, Dekking and Queffélec in the uniformly recurrent case. We show that the same criterion characterizes automaticity of minimal substitutive systems.
Furthermore, we construct a minimal substitutive system whose maximal equicontinuous factor is the 2-adic odometer, and for which the corresponding factor map is everywhere uncountable-to-one. We conjecture that a minimal substitutive system is k-automatic if and only if it is an everywhere finite-to-one extension of a k-adic odometer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_13134 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Automaticity of uniformly recurrent substitutive sequences Krawczyk, Elżbieta Müllner, Clemens Number Theory Combinatorics Dynamical Systems Primary: 11B85, 37B10, 68R15. Secondary: 37A45, 68Q45 We provide a complete characterisation of automaticity of uniformly recurrent substitutive sequences in terms of the incidence matrix of the return substitution of the underlying purely substitutive sequence. This resolves a recent question posed by Allouche, Dekking and Queffélec in the uniformly recurrent case. We show that the same criterion characterizes automaticity of minimal substitutive systems. Furthermore, we construct a minimal substitutive system whose maximal equicontinuous factor is the 2-adic odometer, and for which the corresponding factor map is everywhere uncountable-to-one. We conjecture that a minimal substitutive system is k-automatic if and only if it is an everywhere finite-to-one extension of a k-adic odometer. |
| title | Automaticity of uniformly recurrent substitutive sequences |
| topic | Number Theory Combinatorics Dynamical Systems Primary: 11B85, 37B10, 68R15. Secondary: 37A45, 68Q45 |
| url | https://arxiv.org/abs/2111.13134 |