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

TL;DR

This paper extends the concepts of coverability and multi-scale coverability from one-dimensional words to two-dimensional pictures, comparing their properties and regularities in both contexts.

Contribution

It introduces the extension of quasiperiodic notions to two-dimensional words and compares their regularity properties with one-dimensional cases.

Findings

Multi-scale coverability implies uniform recurrence in 2D

Quasiperiodicity and multi-scale coverability differ in topological entropy

Properties of quasiperiods influence the regularity of quasiperiodic words

Abstract

A word is quasiperiodic (or coverable) if it can be covered with occurrences of another finite word, called its quasiperiod. A word is multi-scale quasiperiodic (or multi-scale coverable) if it has infinitely many different quasiperiods. These notions were previously studied in the domains of text algorithms and combinatorics of right infinite words. We extend them to infinite pictures (two-dimensional words). Then we compare the regularity properties (uniform recurrence, uniform frequencies, topological entropy) of quasiperiodicity with multi-scale quasiperiodicity, and we also compare each of them with its one-dimensional counterpart. We also study which properties of quasiperiods enforce properties on the quasiperiodic words.