Journal of Integer Sequences, Vol. 16 (2013), Article 13.2.5

Some Properties of Abelian Return Words

M. Rigo, P. Salimov, and E. Vandomme
Department of Mathematics
University of Liège
Grande traverse 12 (B37)
B-4000 Liège


We investigate some properties of abelian return words as recently introduced by Puzynina and Zamboni. In particular, we obtain a characterization of Sturmian words with nonzero intercept in terms of the finiteness of the set of abelian return words to all prefixes. We describe this set of abelian returns for the Fibonacci word but also for the 2-automatic Thue-Morse word. We also investigate the relationship existing between abelian complexity and finiteness of the set of abelian returns to all prefixes.

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(Concerned with sequences A001911 A003622 A003849 A010060 A022342.)

Received June 2 2012; revised version received September 24 2012. Published in Journal of Integer Sequences, March 2 2013.

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