Recurrence in Multidimensional Words

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.