Recurrence in Multidimensional Words

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

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

Personalised recommendations

Citepaper 