On extended boundary sequences of morphic and Sturmian words

Generalizing the notion of the boundary sequence introduced by Chen and Wen, the nth term of the -boundary pound sequence of an infinite word is the finite set of pairs (u, v) of prefixes and suffixes of length pound appearing in factors uyv of length n + pound (n pound 1). Otherwise stated, for increasing values of n, one looks for all pairs of factors of length pound separated by n - pound symbols.For the large class of addable abstract numeration systems S, we show that if an infinite word is S-automatic, then the same holds for its -boundary pound sequence. In particular, they are both morphic (or generated by an HD0L system). To precise the limits of this result, we discuss examples of non-addable numeration systems and S automatic words for which the boundary sequence is nevertheless S-automatic and conversely, S-automatic words with a boundary sequence that is not S-automatic. In the second part of the paper, we study the -boundary pound 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.