# Recurrence along directions in multidimensional words

## TL;DR

This paper introduces and explores new notions of uniform recurrence in multidimensional words, focusing on recurrence along all directions, and provides constructions and analyses of such words, including rotation words and fixed points of morphisms.

## Contribution

It defines and studies uniformly recurrent words along all directions in multidimensional words, offering new constructions and analyzing specific classes like rotation words and morphism fixed points.

## Key findings

- Constructed multidimensional words uniformly recurrent along all directions.
- Established relations between recurrence properties and specific word classes.
- Provided examples of words satisfying increasingly strong recurrence conditions.

## 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 $(s_1,\ldots,s_d)\in\mathbb{N}^d$ there exists $n\in\mathbb{N}$ such that each block of size $(n,\ldots,n)$ contains the prefix of size $(s_1,\ldots,s_d)$. We are interested in a modification of this property. Namely, we ask that for each rational direction $(q_1,\ldots,q_d)$, each rectangular prefix occurs along this direction in positions $\ell(q_1,\ldots,q_d)$ 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.