On extended boundary sequences of morphic and Sturmian words

TL;DR

This paper explores the properties of extended boundary sequences of morphic and Sturmian words, demonstrating their automaticity in certain numeration systems and analyzing their structure through morphisms and sliding block codes.

Contribution

It generalizes the boundary sequence concept, establishes automaticity results for $S$-automatic words in addable numeration systems, and characterizes boundary sequences of Sturmian words via morphisms and sliding block codes.

Findings

Boundary sequences are morphic for $S$-automatic words in addable systems.

Examples show the limits of automaticity transfer between words and boundary sequences.

Boundary sequences of Sturmian words relate to characteristic Sturmian words through morphisms.

Abstract

Generalizing the notion of the boundary sequence introduced by Chen and Wen, the $n$th term of the $\ell$-boundary sequence of an infinite word is the finite set of pairs $(u,v)$ of prefixes and suffixes of length $\ell$ appearing in factors $uyv$ of length $n+\ell$ ($n\ge \ell\ge 1$). Otherwise stated, for increasing values of $n$, one looks for all pairs of factors of length $\ell$ separated by $n-\ell$ 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 $\ell$-boundary 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 $\ell$-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.