### Abstract

In this paper we study various modifications of the notion of uniform recurrence in multidimensional infinite words. A d-dimensional infinite word is said to be uniformly recurrent if for each (Formula Presented) there exists (Formula Presented) such that each block of size (Formula Presented) contains the prefix of size (Formula Presented). We introduce and study a new notion of uniform recurrence of multidimensional infinite words: for each rational slope (Formula Presented), each rectangular prefix must occur along this slope, that is in positions (Formula Presented), with bounded gaps. Such words are called uniformly recurrent along all directions. We provide several constructions of multidimensional infinite words satisfying this condition, and more generally, a series of three conditions on recurrence. We study general properties of these new notions and in particular we study the strong uniform recurrence of fixed points of square morphisms.

Original language | English |
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Title of host publication | Language and Automata Theory and Applications - 13th International Conference, LATA 2019, Proceedings |

Editors | Dana Shapira, Alexander Okhotin, Carlos Martín-Vide |

Publisher | Springer |

Pages | 397-408 |

Number of pages | 12 |

ISBN (Print) | 9783030134341 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

Event | 13th International Conference on Language and Automata Theory and Applications, LATA 2019 - St. Petersburg Duration: 26 Mar 2019 → 29 Mar 2019 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 11417 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 13th International Conference on Language and Automata Theory and Applications, LATA 2019 |
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Country | Russian Federation |

City | St. Petersburg |

Period | 26/03/19 → 29/03/19 |

### Fingerprint

### Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Language and Automata Theory and Applications - 13th International Conference, LATA 2019, Proceedings*(pp. 397-408). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11417 LNCS). Springer. https://doi.org/10.1007/978-3-030-13435-8_29

}

*Language and Automata Theory and Applications - 13th International Conference, LATA 2019, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 11417 LNCS, Springer, pp. 397-408, St. Petersburg, 26/03/19. https://doi.org/10.1007/978-3-030-13435-8_29

**Recurrence in Multidimensional Words.** / Charlier, Émilie; Puzynina, Svetlana; Vandomme, Élise.

Research output

TY - GEN

T1 - Recurrence in Multidimensional Words

AU - Charlier, Émilie

AU - Puzynina, Svetlana

AU - Vandomme, Élise

PY - 2019/1/1

Y1 - 2019/1/1

N2 - In this paper we study various modifications of the notion of uniform recurrence in multidimensional infinite words. A d-dimensional infinite word is said to be uniformly recurrent if for each (Formula Presented) there exists (Formula Presented) such that each block of size (Formula Presented) contains the prefix of size (Formula Presented). We introduce and study a new notion of uniform recurrence of multidimensional infinite words: for each rational slope (Formula Presented), each rectangular prefix must occur along this slope, that is in positions (Formula Presented), with bounded gaps. Such words are called uniformly recurrent along all directions. We provide several constructions of multidimensional infinite words satisfying this condition, and more generally, a series of three conditions on recurrence. We study general properties of these new notions and in particular we study the strong uniform recurrence of fixed points of square morphisms.

AB - In this paper we study various modifications of the notion of uniform recurrence in multidimensional infinite words. A d-dimensional infinite word is said to be uniformly recurrent if for each (Formula Presented) there exists (Formula Presented) such that each block of size (Formula Presented) contains the prefix of size (Formula Presented). We introduce and study a new notion of uniform recurrence of multidimensional infinite words: for each rational slope (Formula Presented), each rectangular prefix must occur along this slope, that is in positions (Formula Presented), with bounded gaps. Such words are called uniformly recurrent along all directions. We provide several constructions of multidimensional infinite words satisfying this condition, and more generally, a series of three conditions on recurrence. We study general properties of these new notions and in particular we study the strong uniform recurrence of fixed points of square morphisms.

KW - Multidimensional morphisms

KW - Multidimensional words

KW - Uniform recurrence

UR - http://www.scopus.com/inward/record.url?scp=85064036877&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-13435-8_29

DO - 10.1007/978-3-030-13435-8_29

M3 - Conference contribution

AN - SCOPUS:85064036877

SN - 9783030134341

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 397

EP - 408

BT - Language and Automata Theory and Applications - 13th International Conference, LATA 2019, Proceedings

A2 - Shapira, Dana

A2 - Okhotin, Alexander

A2 - Martín-Vide, Carlos

PB - Springer

ER -