multidimensional word; recurrence along directions; morphism
Abstract :
[en] In this paper we introduce and study new notions of uniform recurrence in multidimensional words. A d-dimensional word is called uniformly recurrent if for all s_1,...,s_d, there exists n such that each block of size (n,…,n) contains the prefix of size (s1,…,sd). We are interested in a modification of this property. Namely, we ask that for each rational direction (q_1,…,q_d), each rectangular prefix occurs along this direction in positions ℓ(q1,…,qd) with bounded gaps. Such words are called uniformly recurrent along all directions. We provide several constructions of multidimensional words satisfying this condition, and more generally, a series of four increasingly stronger conditions. In particular, we study the uniform recurrence along directions of multidimentional rotation words and of fixed points of square morphisms.
Disciplines :
Mathematics
Author, co-author :
Charlier, Emilie ; Université de Liège - ULiège > Département de mathématique > Mathématiques discrètes
Puzynina, Svetlana
Vandomme, Elise ; Université de Liège - ULiège > Département de mathématique > Probabilités et statistique mathématique
Language :
English
Title :
Recurrence along directions in multidimensional words
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