Coverability and multi-scale coverability on infinite pictures

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. •We extend the notion of Quasiperiodicity (coverability) to two-dimensional words.•We show (in)dependence links between Quasiperiodicity and other classical notions.•We study whether this local notion (Quasiperiodicity) may imply global properties.•We do the same for the stronger notion of Multi-scale Quasiperiodicity.