Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Recurrence along directions in multidimensional words   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.

11
The categories of this paper are: hep-lat
19
The categories of this paper are: eess
13
The categories of this paper are: quant-ph
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys
28
The categories of this paper are: alg-geom
21
The categories of this paper are: adap-org
7
The categories of this paper are: hep-th
23
The categories of this paper are: solv-int
5
The categories of this paper are: gr-qc
8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg
24
The categories of this paper are: dg-ga
0
The categories of this paper are: hep-ph
27
The categories of this paper are: comp-gas