Recurrence in Multidimensional Words

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


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.


Uniform recurrence Multidimensional words Multidimensional morphisms 



We are grateful to Mathieu Sablik for inspiring discussions.


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Copyright information

© 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

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