Recurrence along directions in multidimensional words

Abstract

In this paper we introduce and study new notions of uniform recurrence in multidimensional words. A $d$-dimensional word is called \emph{uniformly recurrent} if for all $(s_1,\dots ,s_d)\in {\mathbb{N}}^d$ there exists $n\in \mathbb{N}$ such that each block of size $(n,\dots ,n)$ contains the prefix of size $(s_1,\dots ,s_d)$. We are interested in a modification of this property. Namely, we ask that for each rational direction $(q_1,\dots ,q_d)$, each rectangular prefix occurs along this direction in positions $\ell (q_1,\dots ,q_d)$ with bounded gaps. Such words are called \emph{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.