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 $(s_1,\ldots,s_d)\in\N^d$ there exists $n\in\N$ 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.

中文翻译：

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多维词中沿方向循环

在本文中，我们介绍并研究了多维词中均匀递归的新概念。如果对于所有 $(s_1,\ldots,s_d)\in\N^d$ 存在 $n\in\N$ 使得每个大小为 $ (n,\ldots,n)$ 包含大小为 $(s_1,\ldots,s_d)$ 的前缀。我们有兴趣修改此属性。也就是说，我们要求对于每个有理方向 $(q_1,\ldots,q_d)$，每个矩形前缀都沿着这个方向出现在位置 $\ell(q_1,\ldots,q_d)$ 中，并带有有界间隙。这样的词被称为\emph{沿所有方向均匀循环}。我们提供了满足这个条件的多维词的几种结构，更一般地说，是一系列四个越来越强的条件。特别是，