Coverability in Two Dimensions

Abstract

A word is quasiperiodic (or coverable) if it can be covered by occurrences of another finite word, called its quasiperiod. This notion was previously studied in the domains of text algorithms and combinatorics of right infinite words. We extend several results to two dimensions. We also characterize all rectangular words that cover non-periodic two-dimensional infinite words. Then we focus on two-dimensional words with infinitely many quasiperiods. We show that such words have zero entropy. However, contrarily to the one-dimensional case, they may not be uniformly recurrent.

Periodicity in rectangular arrays

“…are not, as they can be written in the form [1] 2×2 or [1 2] 2×1 respectively. As a consequence of Theorem 3 we get another proof of Lemma 3.3 in [10].…”

“…Marcus and Sokol [19] considered two-dimensional Lyndon words. Crochemore, Iliopoulos, and Korda [8] and, more recently, Gamard and Richomme [10], considered quasiperiodicity in two dimensions. However, with the exception of this latter paper, where Corollary 9 can be found, none of this work is directly related to the problems we consider in this paper.…”

Two-dimensional maximal repetitions

“…The right-angle lattice periodicity is also used by Gamard and Richomme [11] where the primitive roots of 2D arrays are studied. A matrix is defined as primitive if it cannot be broken down to a repeating factor vertically and/or horizontally.…”

Comparison of Coverability and Multi-Scale Coverability in One and Two Dimensions

“…In [7], we continued the study of two-dimensional coverability by generalizing the results from [13] to infinite pictures. In particular, we have shown some dependence and independence results between coverability (resp.…”

“…Our preliminary results (from [7]) are summarized in the following table. Here, ⊥ means "independent", ?…”