Locally Periodic Versus Globally Periodic Infinite Words

We call a one-way infinite word w over a finite alphabet (ρ, l)-repetitive if all long enough prefixes of w contain as a suffix a ρth power (or more generally a repetition of order ρ) 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.