# Recurrence in Multidimensional Words

• Émilie Charlier
• Svetlana Puzynina
• Élise Vandomme
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11417)

## 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.

## Keywords

Uniform recurrence Multidimensional words Multidimensional morphisms

## Notes

### Acknowledgements

We are grateful to Mathieu Sablik for inspiring discussions.

## References

1. 1.
Berthé, V., Vuillon, L.: Tilings and rotations on the torus: a two-dimensional generalization of Sturmian sequences. Discrete Math. 223, 27–53 (2000)
2. 2.
Cassaigne, J.: Double sequences with complexity $$mn+1$$. J. Autom. Lang. Combin. 4, 153–170 (1999)
3. 3.
Cyr, V., Kra, B.: Nonexpansive $$\mathbb{Z}^2$$-subdynamics and Nivat’s conjecture. Trans. Am. Math. Soc. 367(9), 6487–6537 (2015)
4. 4.
Durand, F., Rigo, M.: Multidimensional extension of the Morse-Hedlund theorem. Eur. J. Combin. 34, 391–409 (2013)
5. 5.
Kari, J., Szabados, M.: An algebraic geometric approach to Nivat’s conjecture. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9135, pp. 273–285. Springer, Heidelberg (2015).
6. 6.
Nivat, M.: Invited talk at ICALP, Bologna (1997)Google Scholar
7. 7.
Vuillon, L.: Combinatoire des motifs d’une suite sturmienne bidimensionnelle. Theor. Comput. Sci 209, 261–285 (1998)

© Springer Nature Switzerland AG 2019

## Authors and Affiliations

• Émilie Charlier
• 1
• Svetlana Puzynina
• 2
• 3
Email author
• Élise Vandomme
• 4
1. 1.University of LiegeLiègeBelgium
2. 2.Saint Petersburg State UniversitySaint PetersburgRussia
3. 3.Sobolev Institute of MathematicsNovosibirskRussia
4. 4.Czech Technical University in PraguePragueCzech Republic