## 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

### Keywords:

infinite words; periodicity
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### References:

 [1] Choffrut, C.; Karhumäki, J., Combinatorics of words, (), 329-438 [2] Lothaire, M., Combinatorics on words, (1983), Addison-Wesley Reading · Zbl 0514.20045 [3] Mignosi, F.; Pirillo, G., Repetitions in the Fibonacci infinite word, RAIRO theor. inform. appl., 26, 199-204, (1992) · Zbl 0761.68078 [4] Mignosi, F.; Restivo, A.; Salemi, S., A periodicity theorem on words and applications, Mfcs’95, 969, (1995), Springer Berlin, p. 337-348 · Zbl 1193.68202 [5] Mignosi, F.; Restivo, A.; Salemi, S., Periodicity and Golden ratio, Theoret. comput. sci., 204, 153-167, (1998) · Zbl 0913.68162
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