On the rigidity of Arnoux-Rauzy words
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866908425567338496 |
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| author | Berthé, Valérie Puzynina, Svetlana |
| author_facet | Berthé, Valérie Puzynina, Svetlana |
| contents | An infinite word generated by a substitution is rigid if all the substitutions which fix this word are powers of a same substitution. Sturmian words as well as characteristic Arnoux-Rauzy words are known to be rigid. In the present paper, we prove that all Arnoux-Rauzy words are rigid. The proof relies on two main ingredients: firstly, the fact that the primitive substitutions that fix an Arnoux-Rauzy word share a common power, and secondly, the notion of normal form of an episturmian substitution (i.e., a substitution that fixes an Arnoux-Rauzy word). The main difficulty is then of a combinatorial nature and relies on the normalization process when taking powers of episturmian substitutions: the normal form of a square is not necessarily equal to the square of the normal forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_07707 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the rigidity of Arnoux-Rauzy words Berthé, Valérie Puzynina, Svetlana Discrete Mathematics An infinite word generated by a substitution is rigid if all the substitutions which fix this word are powers of a same substitution. Sturmian words as well as characteristic Arnoux-Rauzy words are known to be rigid. In the present paper, we prove that all Arnoux-Rauzy words are rigid. The proof relies on two main ingredients: firstly, the fact that the primitive substitutions that fix an Arnoux-Rauzy word share a common power, and secondly, the notion of normal form of an episturmian substitution (i.e., a substitution that fixes an Arnoux-Rauzy word). The main difficulty is then of a combinatorial nature and relies on the normalization process when taking powers of episturmian substitutions: the normal form of a square is not necessarily equal to the square of the normal forms. |
| title | On the rigidity of Arnoux-Rauzy words |
| topic | Discrete Mathematics |
| url | https://arxiv.org/abs/2205.07707 |