Infinite special branches in words associated with beta-expansions
A Parry number is a real number β > 1 such that the Rényi β-expansion of 1 is finite or infinite eventually periodic. If this expansion is finite, β is said to be a simple Parry number. Remind that any Pisot number is a Parry number. In a previous work we have determined the complexity of the fix...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2007-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/415/pdf |
| _version_ | 1797270129233362944 |
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| author | Christiane Frougny Zuzana Masáková Edita Pelantová |
| author_facet | Christiane Frougny Zuzana Masáková Edita Pelantová |
| author_sort | Christiane Frougny |
| collection | DOAJ |
| description | A Parry number is a real number β > 1 such that the Rényi β-expansion of 1 is finite or infinite eventually periodic. If this expansion is finite, β is said to be a simple Parry number. Remind that any Pisot number is a Parry number. In a previous work we have determined the complexity of the fixed point uβ of the canonical substitution associated with β-expansions, when β is a simple Parry number. In this paper we consider the case where β is a non-simple Parry number. We determine the structure of infinite left special branches, which are an important tool for the computation of the complexity of uβ. These results allow in particular to obtain the following characterization: the infinite word uβ is Sturmian if and only if β is a quadratic Pisot unit. |
| first_indexed | 2024-04-25T01:59:21Z |
| format | Article |
| id | doaj.art-319fbda4085e4d539ada7e3c94ccbdba |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T01:59:21Z |
| publishDate | 2007-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-319fbda4085e4d539ada7e3c94ccbdba2024-03-07T15:09:28ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502007-01-01Vol. 9 no. 210.46298/dmtcs.415415Infinite special branches in words associated with beta-expansionsChristiane Frougny0Zuzana Masáková1Edita Pelantová2https://orcid.org/0000-0003-3817-2943Laboratoire d'informatique Algorithmique : Fondements et ApplicationsDepartment of Mathematics [Prague]Department of Mathematics [Prague]A Parry number is a real number β > 1 such that the Rényi β-expansion of 1 is finite or infinite eventually periodic. If this expansion is finite, β is said to be a simple Parry number. Remind that any Pisot number is a Parry number. In a previous work we have determined the complexity of the fixed point uβ of the canonical substitution associated with β-expansions, when β is a simple Parry number. In this paper we consider the case where β is a non-simple Parry number. We determine the structure of infinite left special branches, which are an important tool for the computation of the complexity of uβ. These results allow in particular to obtain the following characterization: the infinite word uβ is Sturmian if and only if β is a quadratic Pisot unit.https://dmtcs.episciences.org/415/pdffactor complexity functionparry numbers[info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | Christiane Frougny Zuzana Masáková Edita Pelantová Infinite special branches in words associated with beta-expansions Discrete Mathematics & Theoretical Computer Science factor complexity function parry numbers [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| title | Infinite special branches in words associated with beta-expansions |
| title_full | Infinite special branches in words associated with beta-expansions |
| title_fullStr | Infinite special branches in words associated with beta-expansions |
| title_full_unstemmed | Infinite special branches in words associated with beta-expansions |
| title_short | Infinite special branches in words associated with beta-expansions |
| title_sort | infinite special branches in words associated with beta expansions |
| topic | factor complexity function parry numbers [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/415/pdf |
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