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 \((n_1,\ldots ,n_d)\in \mathbb {N}^d\) there exists \(N\in \mathbb {N}\) such that each block of size \((N,\ldots ,N)\) contains the prefix of size \((n_1,\ldots ,n_d)\). We introduce and study a new notion of uniform recurrence of multidimensional infinite words: for each rational slope \((q_1,\ldots ,q_d)\), each rectangular prefix must occur along this slope, that is in positions \(\ell (q_1,\ldots ,q_d)\), 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.
The second author is partially supported by Russian Foundation of Basic Research (grant 18-31-00118). The last author acknowledges financial support by the Ministry of Education, Youth and Sports of the Czech Republic (project no. CZ.02.1.01/0.0/0.0/16_019/0000778).
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Acknowledgements
We are grateful to Mathieu Sablik for inspiring discussions.
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Charlier, É., Puzynina, S., Vandomme, É. (2019). Recurrence in Multidimensional Words. In: Martín-Vide, C., Okhotin, A., Shapira, D. (eds) Language and Automata Theory and Applications. LATA 2019. Lecture Notes in Computer Science(), vol 11417. Springer, Cham. https://doi.org/10.1007/978-3-030-13435-8_29
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DOI: https://doi.org/10.1007/978-3-030-13435-8_29
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