Recurrence 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 uniformly recurrent if for all ps 1 , . . . , s d q P N d there exists n P N such that each block of size pn, . . . , nq contains the prefix of size ps 1 , . . . , s d q.We are interested in a modification of this property. Namely, we ask that for each rational direction pq 1 , . . . , q d q, each rectangular prefix occurs along this direction in positions pq 1 , . . . , q d q with bounded gaps. Such words are called 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.

Aperiodic Two-Dimensional Words of Small Abelian Complexity

“…A two-dimensional word w is called recurrent if each its factor occurs in it infinitely many times. An infinite word w is called uniformly recurrent if for each integer n there exists an integer N such that each square N × N factor of w contains all square factors of w of size n × n. For different versions of recurrence in 2D-words, we refer to [4].…”