eng
Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-08-22
79:1
79:16
10.4230/LIPIcs.MFCS.2022.79
article
On Extended Boundary Sequences of Morphic and Sturmian Words
Rigo, Michel
1
https://orcid.org/0000-0001-7463-8507
Stipulanti, Manon
1
https://orcid.org/0000-0002-2805-2465
Whiteland, Markus A.
1
https://orcid.org/0000-0002-6006-9902
Department of Mathematics, University of LiΓ¨ge, Belgium
Generalizing the notion of the boundary sequence introduced by Chen and Wen, the nth term of the π-boundary sequence of an infinite word is the finite set of pairs (u,v) of prefixes and suffixes of length π appearing in factors uyv of length n+π (n β₯ π β₯ 1). Otherwise stated, for increasing values of n, one looks for all pairs of factors of length π separated by n-π symbols.
For the large class of addable numeration systems U, we show that if an infinite word is U-automatic, then the same holds for its π-boundary sequence. In particular, they are both morphic (or generated by an HD0L system). We also provide examples of numeration systems and U-automatic words with a boundary sequence that is not U-automatic. In the second part of the paper, we study the π-boundary sequence of a Sturmian word. We show that it is obtained through a sliding block code from the characteristic Sturmian word of the same slope. We also show that it is the image under a morphism of some other characteristic Sturmian word.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol241-mfcs2022/LIPIcs.MFCS.2022.79/LIPIcs.MFCS.2022.79.pdf
Boundary sequences
Sturmian words
Numeration systems
Automata
Graph of addition