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Locally periodic versus globally periodic infinite words. (English) Zbl 1011.68070

Summary: We call a one-way infinite word \(w\) over a finite alphabet \((\rho,l)\)-repetitive if all long enough prefixes of \(w\) contain as a suffix a \(\rho\)th power (or more generally a repetition of order \(\rho\)) of a word of length at most \(l\). We show that each (2,4)-repetitive word is ultimately periodic, as well as that there exist continuum many, and hence also nonultimately periodic, (2,5)-repetitive words. Further, we characterize nonultimately periodic (2,5)-repetitive words both structurally and algebraically.

MSC:

68R15 Combinatorics on words
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