Recurrence along directions in multidimensional words [article]

Émilie Charlier, Svetlana Puzynina, Élise Vandomme
2020 arXiv   pre-print
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)∈N^d there exists n∈N such that each block of size (n,...,n) contains the prefix of size (s_1,...,s_d). 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(q_1,...,q_d) with bounded gaps. Such
more » ... rds 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.
arXiv:1907.00192v2 fatcat:ybjhhw6ylbdylmauguiseyqpvu